Measure theory

Measure theory studies measures, which assign sizes to measurable sets, and the Lebesgue integral, which accumulates values weighted by those sizes. The same framework describes Lebesgue measure on the real numbers, counting measure on discrete collections, and probability measures on possible outcomes. The choice of measure determines what counts as large; the choice of sigma-algebra determines what can be measured.

A striking example separates size from proximity: the rational numbers in $[0,1]$ form a dense set with Lebesgue measure zero, although the whole interval has Lebesgue measure one. This distinction makes almost everywhere statements useful: exceptional null sets can be ignored in a Lebesgue integral even when they meet every open subinterval. The monotone convergence theorem and dominated convergence theorem then give precise conditions for passing a limit through an integral.

A historical starting point is Lebesgue's 1902 thesis, ^[ref]. Its title connects integration with the geometric task of assigning sizes.

A sequence of real measurable functions converges in measure to $f$ when the displayed condition holds for every positive threshold. It controls the measure of the exceptional set at each index, rather than requiring eventual closeness on each sample path. On a probability space it is exactly convergence in probability. On a finite measure space, almost-everywhere convergence implies convergence in measure by continuity of measure on the decreasing tail events. The converse need not hold for the full sequence, but convergence in measure always admits an almost-everywhere convergent subsequence.

Modulo equality almost everywhere, the displayed metric induces convergence in measure and is complete. A sequence Cauchy in measure has a strictly increasing subsequence $f_{n_k}$ with $\mu (|f_{n_{k+1}}-f_{n_k}|>2^{-k})<2^{-k}$. Subadditivity makes the limsup of these exceptional sets null, so the subsequence has summable pointwise successive differences almost everywhere. It therefore has a finite measurable limit. Almost-everywhere convergence implies convergence in measure for finite measure, and the triangle inequality plus the original Cauchy property extends convergence to the full sequence.

For the metric equivalence, $d(f_n,f)\le \epsilon \mu (E)+\mu (|f_n-f|>\epsilon )$ for $0<\epsilon <1$, while $\mu (|f_n-f|>\epsilon )\le d(f_n,f)/\min \{1,\epsilon \}$. The same estimates compare the two Cauchy properties. Define the subsequence limit as zero where no finite limit exists; that exceptional set is measurable by the countable Cauchy criterion, so completeness of the underlying measure is unnecessary.

A subset $A$ of a measurable space $(X,\Sigma )$ is non-measurable when $A\notin \Sigma$. On the real line, unless another sigma-algebra is specified, this commonly means that $A$ is not a Lebesgue measurable set. By completeness of Lebesgue measure, such a set cannot be contained in a Lebesgue null set.

For an example, use the axiom of choice to select one representative from each equivalence class of $[0,1]$ under $x\sim y$ if $x-y\in \mathbb{Q}$, and call the resulting set $V$. The translates $V+q$, for $q\in \mathbb{Q}\cap [-1,1]$, are pairwise disjoint, their union contains $[0,1]$, and their union is contained in $[-1,2]$. If $V$ were a Lebesgue measurable set, translation invariance and countable additivity would make that union have measure zero when $m(V)=0$ and infinite measure when $m(V)>0$. Both conclusions contradict these inclusions. Thus $V$ is non-measurable.

A finite collection of Euclidean open balls has a disjoint subcollection whose concentric triples cover the original union. Greedily retain a largest remaining ball and discard all intersecting ones. Each discarded ball has radius no greater than the retained one and lies in its triple. Lebesgue measure consequently bounds the original union by $3^d$ times the sum of the retained volumes. This covering lemma is distinct from inverse-closedness results for the Wiener algebra.

Every open set of finite Lebesgue measure in Euclidean space is covered, up to a null set, by countably many pairwise disjoint open balls contained in it. At each stage cover a compact subset of the open remainder, choose a packing with the Wiener covering lemma, and remove its closed balls. A fixed positive fraction of the remaining measure is removed each time, while sphere boundaries are null.

Geometric measure theory uses measures and analysis to study geometric objects that need not be smooth. Its measuring rules can distinguish a curve from its surrounding plane or capture the fine structure of a fractal. Hausdorff measure supplies size adapted to an exponent, while Hausdorff dimension identifies the exponent at which that size changes character.

The Cantor set is an elementary starting point: it has zero Lebesgue measure but positive Hausdorff dimension. In Hausdorff dimension of the Cantor set, explicit small covers give an upper bound and a distributed probability measure supplies the matching lower bound. This combination of geometric coverings and analytic control is useful even before more advanced tools for irregular surfaces are introduced.

For a scalar smooth map $\Phi :{\mathbb{R}}^n\to \mathbb{R}$, integration can be decomposed into its level surfaces with the gradient weight displayed in the title. Near a regular level, using $\Phi$ as the normal coordinate gives the volume element $dx=dSds/|\nabla \Phi |$, which explains the formula by integration by substitution. In particular, $\int W(x)\delta (\Phi (x))dx={\int }_{\Phi =0}W/|\nabla \Phi |dS$ when the zero level is regular and the integrals are well-defined. General versions allow Lipschitz functions and maps into higher-dimensional spaces.

Hausdorff measure measures the geometric size of a set using covers by increasingly small pieces. The exponent $s$ chooses the measuring scale: $s=1$ is length-like, $s=2$ is area-like, and noninteger exponents can measure fractals. Unlike a count of occupied grid cells, the construction allows pieces of different sizes and considers every admissible cover.

For $s>0$ and a subset $E$ of a metric space, use the diameter-cost conventionThe first quantity is scale-restricted Hausdorff content. Shrinking $\delta$ removes options from the infimum, so these contents increase to a limit, possibly infinite. The resulting outer measure restricts to a measure on the Borel sets. For $s=0$, counting each nonempty covering piece with cost one gives counting measure.

A different common convention multiplies these quantities by $c_s={\pi }^{s/2}/(2^s\Gamma (1+s/2))$, using the Gamma function. At integer $s$, that normalization agrees with ordinary Euclidean $s$-dimensional volume. Here $c_1=1$, so both conventions make ${\mathcal{H}}^1$ agree with arc length on smooth curves in Euclidean space. Positive normalization constants never change Hausdorff dimension. A definition must specify this constant when numerical values of Hausdorff measure matter.

For a line segment of length $L$, splitting it into $n$ equal pieces gives the cover costIt tends to zero for $s>1$, stays at $L$ for $s=1$, and grows for $s<1$. The matching lower bound comes from the fact that any interval cover must have total length at least $L$. The exponent one is therefore the transition between a measuring rule too coarse to see the segment and one that assigns it infinite size. For a Cantor set, the transition is instead at $\log 2/\log 3$, as shown in Hausdorff dimension of the Cantor set.

This makes Hausdorff measure useful when ordinary Lebesgue measure sees too little: a curve in the plane has zero planar area but can have positive arc length, and the Cantor set has zero line length but positive geometric size at its own dimension. It also separates local covering estimates from global cardinality: a countable dense set has zero Hausdorff dimension, despite meeting every open region of its closure.

An original source for the construction is Felix Hausdorff's ^[ref]. The modern covering and Hausdorff dimension framework is developed in Bishop and Peres, Fractals in Probability and Analysis, §1.2.

For $\delta >0$, scale-restricted Hausdorff content is the infimum of ${\sum }_j(\mathrm{diam}{U}_j{)}^s$ over countable covers of $F$ by sets of diameter at most $\delta$, using the convention in Hausdorff measure. It increases as $\delta$ decreases; its limit is Hausdorff measure. Unrestricted Hausdorff content imposes no diameter bound and can behave differently.

Any explicit cover supplies an upper bound on scale-restricted Hausdorff content, rather than its exact value. The level-$n$ intervals of the Cantor set, for example, giveIf this right-hand side tends to zero, then ${\mathcal{H}}^s(C)=0$. If it grows, that alone proves no positive lower bound: a different cover might be cheaper. The mass distribution principle addresses precisely this gap by bounding the cost of every sufficiently small cover.

The Hausdorff dimension is the critical exponent at which Hausdorff measure changes from infinite to zero:with the standard endpoint convention for an empty supremum. This measures fine-scale geometric size rather than the number of coordinates used to describe the set. A line segment has Hausdorff dimension one; a square has dimension two; the middle-third Cantor set has dimension $\log 2/\log 3\approx 0.63093$.

Why can a dimension be fractional? At a scale reduced by a factor of three, the Cantor set needs twice as many construction intervals. The balance $2(1/3{)}^d=1$ gives $d=\log 2/\log 3$, between the growth rates of finitely many points and a whole interval. Making this argument rigorous requires a cover upper bound and the mass distribution principle lower bound, as in Hausdorff dimension of the Cantor set.

At the critical exponent itself, Hausdorff measure can be zero, finite and positive, or infinite; dimension does not specify that endpoint value. For example, finite sets and countably infinite sets both have dimension zero, although their counting measures differ. Dimension is also different from a finite-resolution estimate: box-counting dimension is another asymptotic definition, and fitting a scaling slope over a handful of observed scales does not establish either dimension.

The metric matters. Multiplying every distance by a fixed positive constant multiplies ${\mathcal{H}}^s$ by that constant to the power $s$ and preserves Hausdorff dimension. A Lipschitz map cannot increase it, because mapped cover diameters increase by at most a fixed factor. A bi-Lipschitz map therefore preserves it by applying this fact in both directions. These properties make Hausdorff dimension useful for comparing geometric models while controlling how their measurements change.

The mass distribution principle converts a diffuse measure into a lower bound on Hausdorff dimension. Let $s>0$ and let a Borel probability measure $\mu$ give a Borel set $E$ full mass. If, for some $C,r_0>0$, every ball centered in $E$ satisfiesthen ${\mathcal{H}}^s(E)\ge 1/C>0$ in the diameter-cost convention, and consequently ${\dim }_{H}E\ge s$.

The idea is to keep unit mass from being crowded into cheap covering pieces. For a piece $U$ of positive diameter $r$ meeting $E$, choose $x\in U\cap E$. A ball of radius $r+\epsilon$ contains $U$; letting $\epsilon \downarrow 0$ gives ${\mu }^{\ast }(U)\le C{r}^s$, where ${\mu }^{\ast }$ is the associated outer measure. Zero-diameter pieces have zero mass by the same estimate. For any sufficiently small cover,Thus every cover costs at least $1/C$. This is the lower-bound counterpart to the explicit covers in scale-restricted Hausdorff content. The Cantor distribution supplies the needed mass in Hausdorff dimension of the Cantor set. ^[ref].

The middle-third Cantor set has Hausdorff dimensionIts Lebesgue measure is zero, yet $0<{\mathcal{H}}^d(C)<\infty$. These assertions measure different aspects of size: interval length is too coarse to distinguish this set from a null set, while the critical fractional measuring rule captures its remaining structure.

At level $n$, the construction covers $C$ by $2^n$ intervals of diameter $3^{-n}$. Their total $s$-cost isFor $s>d$ it tends to zero, proving the upper dimension bound. At $s=d$ it equals one, giving ${\mathcal{H}}^d(C)\le 1$. For $s<d$ this particular cover's cost grows, but that alone would not prove a lower bound on Hausdorff measure.

For the matching lower bound, use the Cantor distribution, which assigns mass $2^{-n}$ to each level-$n$ Cantor cylinder. An interval of length $r$ with $3^{-(n+1)}<r\le 3^{-n}$ meets at most two level-$n$ construction intervals, since their gaps are at least $3^{-n}$. Its mass is therefore at mostEvery covering set of diameter $r$ in the line lies in an interval of length $r$, so any small cover must have total $d$-cost at least $1/4$. This is the mass distribution principle in its interval form: $1/4\le {\mathcal{H}}^d(C)\le 1$, sufficient to prove the claimed dimension. The constants are convenient bounds, rather than an assertion of the exact critical Hausdorff measure. In the diagram, the right panel is a finite-stage approximation to the Cantor function, showing how probability is shared while the middle third receives none.

The reusable pattern is to obtain an upper bound by constructing covers and a lower bound by distributing mass. The same strategy underlies Cartesian powers of the Cantor set and the dimension formula for a finite similarity iterated function system satisfying the open set condition.

Suppose shrinking interval covers of $F$ have length-power sums with logarithmic growth rate $P(\gamma )$, and $P$ is strictly decreasing with zero ${\gamma }_{\ast }$. For every $\gamma >{\gamma }_{\ast }$ those sums tend to zero exponentially. The definition of Hausdorff measure then gives ${\mathcal{H}}^{\gamma }(F)=0$, hence the displayed upper bound. A matching lower bound requires additional information and does not follow from this covering argument alone.

A connected subset $C$ of Euclidean space containing distinct points $a,b$ has Hausdorff dimension at least one. The Lipschitz map $x\mapsto |x-a|$ maps $C$ onto a set containing $[0,|b-a|]$, since continuous images of connected spaces are connected. A Lipschitz map cannot increase Hausdorff dimension, whereas an interval has Hausdorff dimension one. Consequently any set of Hausdorff dimension less than one is totally disconnected.

A finite-dimensional vector measure has scalar measures as components and a total variation norm of a measure defined using their joint vector norm. Finite variation controls every pairing with a bounded continuous vector test. Derivatives in the bounded-variation space are examples.

A vector Radon measure has Radon measures as components. The vector form of the Riesz-Markov-Kakutani representation theorem identifies finite such measures with bounded functionals on $C_0(\Omega ;{\mathbb{R}}^n)$.

Each component of a finite vector measure is absolutely continuous with respect to its variation measure. The Radon-Nikodym theorem gives a vector density $\sigma$. Taking the variation measure of this representation gives $|\mu |=|\sigma ||\mu |$, hence the unit-length property. For a BV derivative this separates the magnitude of variation from its local direction.

A probability space whose measurable space is isomorphic to a Borel subset of a complete separable metric space with its Borel sigma-algebra. This supplies a countable generating algebra and permits the disintegration theorem for a probability measure. Completions can be handled modulo null sets; conditional kernels are chosen on the underlying Borel model.

A measurable space is a set equipped with a sigma-algebra of measurable subsets. A measure space additionally specifies a measure on that sigma-algebra.

A standard Borel space is a measurable space isomorphic to a Borel set of a Polish space, with its Borel sigma-algebra. This measurable regularity permits the usual constructions of random samples and random counting measures, without requiring a particular compatible topology.

A measurable set $N$ is a null set for a measure $\mu$ when $\mu (N)=0$. A property holds almost everywhere when its exceptions lie in a null set. With a complete measure, every subset of a null set is measurable and is itself a null set.

A Lebesgue null set has zero Lebesgue measure. Since Lebesgue measure is complete, every subset of one is also measurable and null. The Cantor set is an uncountable example with positive Hausdorff dimension; the rational numbers form a countable, dense example with Hausdorff dimension zero. Being null does not imply being empty or having finitely many points.

A Borel set of Lebesgue measure zero. Every Lebesgue-null set is contained in a Borel null set, by outer regularity; consequently testing all Borel null sets suffices in the definition of a Sierpiński set.

A random measure assigns a measure ${\mu }_{\omega }$ to each outcome so that $\omega \mapsto {\mu }_{\omega }(A)$ is a random variable for every measurable $A$. A finite-variation process gives pathwise signed random measures through Lebesgue–Stieltjes integration.

A countable measurable partition is a countable family of pairwise disjoint measurable sets whose union has full measure. Its members are its atoms; null atoms can be discarded in statements made almost everywhere.

A partition atom is a member of a measurable partition. For a finite or countable partition, each cell is an atom of a sigma-algebra generated by that partition, whose measurable sets are unions of cells. A partition atom need not be an atom of a measure: an interval cell with positive Lebesgue measure can be split into smaller positive-measure sets that lie outside the generated partition sigma-algebra.

For a countable measurable partition, its entropy is $H_{\mu }(\xi )=-{\sum }_{A\in \xi }\mu (A)\log \mu (A)$, with $0\log 0=0$. This extends entropy of a finite measurable partition and is allowed to be infinite.

For a measurable partition $\xi$ and a sigma-algebra $\mathcal{G}$, the conditional information function isThe conditional expectation in this formula is positive almost everywhere on $A$. Without conditioning, the information is $I_{\mu }(\xi )(x)=-\log \mu (\xi (x))$.

For increasing sigma-algebras ${\mathcal{G}}_n$, put $f_n=I_{\mu }(\xi \mid {\mathcal{G}}_n)$ and $F={\sup }_n{f}_n$. For each atom $A\in \xi$ and $t\ge 0$,Together with the bound by $\mu (A)$ and the tail integral formula for moments, this yields $\int Fd\mu \le H_{\mu }(\xi )+1$. The inequality follows by stopping the conditional-expectation martingale $\mathbb{E}[1_A\mid {\mathcal{G}}_n]$ the first time it falls below $e^{-t}$.

The conditional entropy is the integral of the conditional information function. Its chain rule expresses the entropy of a join of measurable partitions as unconditional entropy plus conditional entropy. Conditioning reduces entropy, also for countable partitions of finite entropy.

The join is the common refinement of two measurable partitions, with atoms given by their nonempty intersections. For a measure-preserving transformation, the block partition is ${\xi }_0^{n-1}={\bigvee }_{j=0}^{n-1}{T}^{-j}\xi$.

A Boolean algebra of subsets contains the empty set and is closed under finite unions and complements. It is consequently closed under finite intersections, differences, and symmetric differences.

A semiring of sets contains the empty set, is closed under intersections, and has the property that the difference of two members is a finite disjoint union of members. Predictable rectangles form a semiring generating the predictable sigma-algebra.

A monotone class of sets is closed under increasing countable unions and decreasing countable intersections.

The monotone class theorem says that the smallest monotone class containing an algebra of sets equals the sigma-algebra generated by that algebra. It extends identities proved on a generating algebra to the full sigma-algebra.

A vector space of bounded functions containing constants and indicators of a generating pi-system, and closed under uniformly bounded monotone pointwise limits, contains all bounded functions measurable for the generated sigma-algebra. This function version extends identities established on elementary integrands to every bounded measurable integrand.

// Target: probability-and-statistics.bigb

A function $f:(X,\mathcal{A})\to (Y,\mathcal{B})$ between measurable spaces is measurable when $f^{-1}(B)\in \mathcal{A}$ for every $B\in \mathcal{B}$.

A Lebesgue measurable function $f:\mathbb{R}\to \mathbb{R}$ has a Lebesgue measurable set as the preimage of every open subset of $\mathbb{R}$. Equivalently, use every Borel set in the target: the sets whose preimages are measurable form a sigma-algebra containing all open sets. The source carries the Lebesgue sigma-algebra and the target its Borel sigma-algebra. A Borel measurable function on $\mathbb{R}$ instead requires the preimages to be Borel in the source as well. Restricting the source to a Lebesgue measurable set uses its induced sigma-algebra.

For real random variables $X,Z$, if $Z$ is measurable with respect to the sigma-algebra generated by a random variable $X$, there is a Borel measurable function $g:\mathbb{R}\to \mathbb{R}$ with $Z=g(X)$. To prove the bounded case, approximate $Z$ pointwise by finite-valued simple functions $Z_m$. Each level set is $X^{-1}(B)$ for a Borel set $B$, so each approximation factors as $Z_m=g_m(X)$ with $g_m$ a Borel measurable function. Apply scalar clipping to $g_m$ with a common bounded interval and use its limit superior to obtain $g$. For unbounded finite $Z$, apply the bounded case to the one-to-one bounded continuous function $t\mapsto \arctan t$, then invert on $(-\pi /2,\pi /2)$ and assign arbitrary values elsewhere. Exact measurability gives exact factorization; assumptions after adjoining null sets give an almost surely equal version.

The essential range of a real or complex measurable function $m$ consists of the values $z$ whose every neighborhood receives inputs of positive measure:It depends on the measure as well as the function. This definition concerns values arbitrarily near $z$, not necessarily inputs at which $m=z$ exactly. It therefore retains meaningful limiting values while ignoring values occurring only on null sets. ^[ref].

For $m(x)=x$ on $(0,1)$ with Lebesgue measure, the ordinary image of a function is $(0,1)$, but the essential range is $[0,1]$. Neither endpoint is attained, yet inputs with $|m(x)|<\epsilon$ or $|m(x)-1|<\epsilon$ have positive measure for every $\epsilon >0$. Every interior value also belongs to the essential range, although its exact fiber is a singleton of zero measure. Positive measure of a fiber is sufficient, but is not necessary, for a value to belong to the essential range.

The two-sided example $g$ from essential infimum and essential supremum illustrates what disappears. It follows $g(x)=x$ on $[0,1]$, except $g(1/4)=-3$ and $g(3/4)=4$. Its ordinary image of a function is $([0,1]\setminus \{1/4,3/4\})\cup \{-3,4\}$, whereas its essential range remains $[0,1]$. The missing ordinary values $1/4$ and $3/4$ still have arbitrarily close neighbors with positive measure; the exceptional values $-3$ and $4$ can each be isolated by a sufficiently small neighborhood whose preimage is a null set.

For the three-point example in L-infinity norm, the values are $1,3,10$. With weights $(0.5,0.5,0)$ the essential range is $\{1,3\}$; with weights $(0.499,0.5,0.001)$ it is $\{1,3,10\}$. A small but positive probability retains ten, so an essential range does not discard a value merely because it is rare. For a real function on a space of positive total measure, the essential infimum and essential supremum are the lower and upper edges of its essential range, allowing infinite bounds. The L-infinity norm instead takes the upper edge of the essential range of $|m|$.

A useful operator connection is the spectrum of a real multiplication operator: on a nonzero L2 space over a sigma-finite measure, multiplication by a real measurable function $m$ has spectrum equal to its essential range, with the natural domain when $m$ is unbounded. Outside that range, $|m-z|$ has a positive lower bound almost everywhere, so multiplication by $1/(m-z)$ gives an inverse function that is a bounded linear operator. Near an essential value, this lower bound fails. Thus discarding null sets is precisely what makes the range match an operator acting on almost everywhere equivalence classes. ^[ref].

Measurability is the property that inverse images of measurable target sets are measurable in the source space. For a stochastic process, joint measurability in time and sample is a different condition from the separate adaptedness condition at each fixed time. The predictable sigma-algebra and progressive measurability encode particular joint measurability requirements.

For topological spaces $X,Y$, a Borel measurable function $f:X\to Y$ is a measurable function for the Borel sigma-algebras on both spaces. Equivalently, the preimage of every open set in $Y$ is a Borel set in $X$. With a different source sigma-algebra $\mathcal{A}$, the phrase measurable into $Y$ with its Borel sigma-algebra means $f^{-1}(B)\in \mathcal{A}$ for every Borel $B\subseteq Y$; specifying the source is essential.

If $f:(X,\mathcal{A})\to (Y,\mathcal{B})$ and $g:(Y,\mathcal{B})\to (Z,\mathcal{C})$ are measurable, then $g\circ f$ is measurable because

If $g$ is a Lebesgue measurable function and $h$ is a Borel measurable function, then $h\circ g$ is a Lebesgue measurable function. For an open set $U$, $h^{-1}(U)$ is a Borel set. By measurability of $g$, $g^{-1}(h^{-1}(U))$ is a Lebesgue measurable set, and it equals $(h\circ g{)}^{-1}(U)$. This is composition of measurable functions with the intermediate Borel sigma-algebra stated explicitly. Merely knowing both functions are Lebesgue measurable would not ensure that the intermediate sigma-algebras match.

For a measurable set $A\subseteq {\mathbb{R}}^n$, its density at $x$ with respect to a locally finite measure $\mu$ iswhen this limit exists.

The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

For a finite singular Borel measure, choose compact subsets of a Lebesgue-null carrier whose omitted mass is arbitrarily small. Outside any such compact subset, sufficiently small balls see only the omitted measure. The uncentered maximal weak-type inequality bounds the volume of points where the omitted measure has large density. Letting its total mass tend to zero shows that the upper spherical density is zero almost everywhere. This argument does not require the null carrier itself to be closed or separated from every point outside it.

If $f$ is locally Lebesgue integrable on ${\mathbb{R}}^n$, then for Lebesgue almost every $x$,

Let $a_n=2^{-2^n}$ and, in angular coordinates near zero, take a bounded indicator alternating between one and zero on the rings $a_{n+1}<|t|\le a_n$. Since $a_{n+1}/a_n\to 0$, the mean over $[-a_n,a_n]$ approaches the value of the outermost ring. Its even and odd subsequences approach one and zero. Thus an integrable function can fail to have a limiting centered average at a specified point, although Lebesgue differentiation theorem ensures convergence almost everywhere.

A point $x\in {\mathbb{R}}^n$ is a Lebesgue point of a locally Lebesgue integrable function $f$ whenThe Lebesgue differentiation theorem says that almost every point is a Lebesgue point.

If $g$ is locally Lebesgue integrable andthen $G^{\prime }(x)=g(x)$ at every Lebesgue point of $g$, and hence almost everywhere.

For every Lebesgue measurable set $A\subseteq {\mathbb{R}}^n$, its Lebesgue density is $1$ at almost every point of $A$ and $0$ at almost every point of its complement.

A point at which a measurable set occupies an asymptotically full fraction of small balls, as expressed above with Lebesgue measure. The Lebesgue density theorem says that almost every point of a measurable set is a density point. On the real line, the same limiting conclusion holds for arbitrary shrinking intervals containing that point, since each interval lies in a centered interval of at most twice its length.

If a measurable set $E\subseteq \mathbb{R}$ has positive measure, then for every $\epsilon >0$ there is a bounded interval $I$ such thatChoose a density-one point of a finite positive-measure portion of $E$ using the Lebesgue density theorem, and take a sufficiently small interval centred there.

A subset $E\subseteq \mathbb{R}$ is a Lebesgue measurable set when it satisfies Carathéodory's criterion for Lebesgue outer measure: for every subset $A\subseteq \mathbb{R}$,These sets form the Lebesgue sigma-algebra, the completion of the Borel sigma-algebra with respect to Lebesgue measure. Equivalently, $E$ differs from a Borel set by a subset of a Borel null set.

The sigma-algebra of all Lebesgue measurable sets, equivalently the completion of a measure of the Borel sigma-algebra under Lebesgue measure. It contains every subset of every Lebesgue null set. Applying Carathéodory's criterion to Lebesgue outer measure gives another construction of this same sigma-algebra.

Every Lebesgue measurable set of finite measure can be approximated arbitrarily closely in measure from outside by an open set and from inside by a compact set. On a bounded interval it can consequently be approximated in symmetric-difference measure by a finite union of intervals.

A property holds almost everywhere with respect to a measure when the set of points where it fails has measure zero.

A measurable function $f$ is Lebesgue integrable when $\int |f|d\lambda <\infty$.