Measure theory

Measure theory supplies a rigorous language for size and integration on general spaces.

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.

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.

Hausdorff measure measures a set by the small-scale cost of covers, using the $s$th powers of their diameters and a standard normalization. In Euclidean dimension one, ${\mathcal{H}}^1$ agrees with arc length on smooth curves; in particular a line segment has its usual length.

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

The essential range of a measurable function $m$ is the set of scalars $z$ for which $\mu (\{x:|m(x)-z|<\epsilon \})>0$ for every $\epsilon >0$. It depends on the measure as well as the function. On a nonzero $L^2$ space over a sigma-finite measure, it describes the spectrum of a real multiplication operator when $m$ is real.

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.

A function into a topological space is Borel measurable when the inverse image of every Borel set is measurable. Equivalently, it is measurable when the target carries its Borel sigma-algebra.

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

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 $x$,

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.

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 Lebesgue measurable set is a member of the completion of the Borel sigma-algebra with respect to Lebesgue measure. In particular, it differs from a Borel set by a null set.

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 $\int |f|d\lambda <\infty$.

A real or complex measurable function is integrable for a measure $\mu$ when its absolute value has finite integral. For a random variable, this is integrability with respect to its probability measure, and makes its expectation finite.

For an integrable random variable $X$ and a sub-sigma-algebra $\mathcal{G}$, the conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$ is the almost-everywhere unique $\mathcal{G}$-measurable integrable random variable satisfyingfor every $G\in \mathcal{G}$.

For nonnegative integrable $f$, the finite measure $A\mapsto {\int }_{A}fd\mu$ restricted to a sub-sigma-algebra ${\Sigma }_0$ is absolutely continuous with respect to the restricted base measure. Its Radon-Nikodym derivative is ${\Sigma }_0$-measurable and has the required integral identities. Positive and negative parts extend the construction to real integrable functions; real and imaginary parts extend it to complex functions. The result is unique almost everywhere.

If $Y$ is a real square-integrable random variable and $X=\mathbb{E}[Y\mid \mathcal{G}]$, then $\mathbb{E}[XY]=\mathbb{E}{X}^2$ and the displayed identity follows by expansion. Thus equality of the second moments forces $X=Y$ as an almost sure equality. This is the equality case of the conditional Jensen inequality for the strictly convex function $z\mapsto z^2$.

Disintegrate over a factor map and take the $L^2$ norm using the conditional probability on each fiber. Its square integrates to the global squared norm. The uniform conditional L2 norm takes the essential supremum of these fiber norms.

For invertible probability systems linked by a factor map, invariance and uniqueness of disintegration identify the pushforward of each conditional measure with the conditional measure on the transformed fiber. The displayed norm identity follows for all integer times on a common full-measure set. It transports uniform conditional error bounds along whole orbits.

The essential supremum over the base of the conditional L2 norm. It may be infinite for a globally square-integrable function. For probability fibers, $\Vert f{\Vert }_2\le \Vert f{\Vert }_{2,\infty \mid Y}$; the uniform norm controls all fibers outside a null set rather than only their average.

For an integrable random variable $X$, a sub-sigma-algebra $\mathcal{G}$, and a convex function $\varphi$ with suitable integrability, $\varphi (\mathbb{E}[X\mid \mathcal{G}])\le \mathbb{E}[\varphi (X)\mid \mathcal{G}]$ almost surely. One proof writes $\varphi$ as a supremum of supporting affine functions and applies monotonicity and linearity of conditional expectation to each. This proves the convex-loss form of the Rao-Blackwell theorem.

For an integrable random variable $Y$, the conditional expectation satisfies $|\mathbb{E}[Y\mid \mathcal{G}]|\le \mathbb{E}[|Y|\mid \mathcal{G}]$ almost surely. Taking expected values gives the L1 contraction of conditional expectation. Applying it to $Y_n-Y$ shows that convergence in L1 is preserved by conditional expectation with respect to any fixed sigma-algebra.

For independent and identically distributed random variables $X_j$ that are integrable random variables, with $S_n={\sum }_{j=1}^n{X}_j$, symmetry gives $\mathbb{E}[X_1\mid S_n,S_{n+1},\dots ]=S_n/n$. The decreasing sigma-algebras generated by these future sums permit application of the reverse martingale convergence theorem.

Taking expectation after conditioning recovers the original expectation. More generally, if $\mathcal{H}\subseteq \mathcal{G}$, then

If sub-sigma-algebras $\mathcal{G}$ and $\mathcal{H}$ are independent, every event in $\mathcal{G}\cap \mathcal{H}$ is independent of itself and consequently has probability zero or one. Thusalmost surely for every integrable random variable $X$.

In general,Equality does hold when either sigma-algebra contains the other or when they are independent.

For nonnegative random variables $X_n$ and a sigma-algebra $\mathcal{G}$,almost surely. It follows from the ordinary Fatou lemma after testing both sides on every event in $\mathcal{G}$.

For nonnegative measurable functions,

For nonnegative numbers $a_{n,k}$, Fatou lemma applied to counting measure gives

SetThen $0\le g_n\uparrow {\mathrm{liminf}}_k{f}_k$. The monotone convergence theorem and $g_n\le f_k$ for every $k\ge n$ give

On $([0,1],\lambda )$, the functionsconverge pointwise to zero but satisfy $\int f_{n}d\lambda =1$. Hence the two sides of Fatou lemma can be zero and one.

Ergodic theory studies the long-time statistical behaviour of measure-preserving transformations.

For a measurable transformation $T$, a measure is invariant if the pushforward measure equals the original measure, or equivalently $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. This is the measure used in a measure-preserving system.

A Borel probability measure is invariant for a Markov kernel semigroup if every transition preserves it. For a Feller semigroup on a compact metric space, it suffices to test the displayed identity on continuous functions. Kernel Jensen inequality gives contraction on $L^p(\mu )$, and sup-norm strong continuity followed by density gives strong continuity there for $1\le p<\infty$.

For a Feller semigroup on a compact metric space, time-averaged transition laws have weakly convergent subsequences by compactness of probability measures on a compact metric space. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.

Cutting and stacking constructs measure-preserving transformations from towers of equal-measure levels. Cut a tower into equal-width subcolumns, add new spacer levels, and stack the resulting columns. Map each level to the one above by a measure-preserving identification; subsequent stages extend the partial map. When uncovered and top-level measures tend to zero, the limiting map is defined modulo null sets.

The standard three-cut Chacon transformation is an invertible measure-preserving transformation built by cutting and stacking with spacer vector $(0,1,0)$ at every stage. Cut the current tower into three equal subcolumns, place one spacer above the middle, and stack left to right. With initial width $2/3$ and height one on a unit interval, level counts and widths are $h_j=(3^{j+1}-1)/2$ and $w_j=2/3^{j+1}$. Tower measure tends to one, using total spacer measure $1/3$. Tower words obey $W_{j+1}=W_j{W}_{j}1W_j$.

A continuous self-map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure. An irrational rotation of the circle is uniquely ergodic: invariance multiplies its nonzero-index Fourier coefficients by nontrivial phases, forcing those coefficients to vanish. The Stone-Weierstrass theorem then identifies the invariant measure as normalized Lebesgue measure.

For a uniquely ergodic continuous map $T$ of a compact metric space, every continuous $f$ satisfies $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to \int fd\mu$ uniformly in $x$. Otherwise choose increasingly long orbit empirical measures with a fixed discrepancy. Compactness provides a weak limit; the telescoping identity makes it invariant, so it must be the unique measure $\mu$, contradicting that discrepancy. Compactness and continuity are essential hypotheses.

For a bounded sequence $(u_n)$ in a Hilbert space, put $c_h={\mathrm{limsup}}_{N\to \infty }|N^{-1}{\sum }_{n=0}^{N-h-1}⟨u_{n+h},u_n⟩|$. If $H^{-1}{\sum }_{h=1}^H{c}_h\to 0$, then $N^{-1}{\sum }_{n=0}^{N-1}{u}_n\to 0$ in norm. Since the $c_h$ are bounded, it suffices that $(c_h)$ has convergence in density of a sequence to zero. The proof applies the Cauchy-Schwarz inequality to an average of shifted sums.

For complex numbers $|u_n|\le 1$, $0\le n<N$, and $1\le H\le N$, put $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. ThenExtend the sequence by zero, count each summand in $H$ consecutive windows, and apply the Cauchy-Schwarz inequality to their sum. Expansion of the squared window sums gives the stated coefficients. Fixing $H$ and making each normalized correlation tend to zero bounds the limiting squared average by $1/H$; then let $H$ tend to infinity. This is the scalar finite form underlying the Van der Corput lemma.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is equidistributed if the limiting frequency in every half-open interval equals that interval's length. Equivalently, its empirical probability measures converge to normalized Lebesgue measure when tested against every continuous function.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is an equidistributed sequence if and only if $N^{-1}{\sum }_{n=0}^{N-1}{e}^{2\pi im{x}_n}\to 0$ for every nonzero integer $m$. Approximation of continuous functions by trigonometric polynomials proves the criterion.

If a circle-valued sequence is not an equidistributed sequence, some positive-shift difference is not equidistributed either. The contrapositive follows by applying the Van der Corput inequality for finite scalar sequences to every nonzero integer exponential sum. Thus cancellation for all fixed nonzero differences implies equidistribution of the original sequence.

If $a$ is irrational and $b,c$ are real, the fractional parts of $a{n}^2+bn+c$ form an equidistributed sequence. In the Van der Corput lemma, every nonzero-lag correlation of $e^{2\pi im(a{n}^2+bn+c)}$ is a geometric series with irrational frequency $2mah$ and tends to zero. The Weyl criterion then applies.

A measurable transformation $T:X\to X$ is nonsingular with respect to $\mu$ when $\mu (T^{-1}N)=0$ for every $\mu$-null set $N$, equivalently when the pushforward measure $T_{\ast }\mu$ is absolutely continuous with respect to $\mu$.

For a measurable dynamical map $F$, the Koopman operator acts on observables by $K_{F}g=g\circ F$. If $F$ is nonsingular, it defines an operator on equivalence classes of measurable functions.

An $L^2$ observable of an invertible probability measure-preserving system is almost periodic when its Koopman operator orbit is totally bounded in the global $L^2$ norm. A noninvertible system uses its forward orbit. This is the Hilbert-space dynamical definition, distinguished from uniform almost periodicity for continuous functions of a real variable.

For a probability measure-preserving system, if $⟨U_T^{n}f,f⟩\to |\int fd\mu {|}^2$, then $U_T^{n}f$ converges weakly in $L^2$ to $(\int fd\mu )1$. Centre $v=f-(\int f)1$; its autocorrelations tend to zero. The isometry identity gives the same decay against each $U_T^{k}v$, hence by approximation against their closed forward span. Against its orthogonal complement the correlations are exactly zero. Orthogonal decomposition by a closed subspace finishes the proof without invertibility.

For a nonsingular map $F$ on a probability space, $K_F$ extends boundedly to $L^2$ exactly when $d(F_{\ast }\mu )/d\mu \in L^{\infty }$. In that case

A measurable transformation $T:X\to X$ preserves a measure $\mu$ when $\mu (T^{-1}A)=\mu (A)$ for every measurable set $A$.

Such a measurable self-map preserves Lebesgue measure. On the unit interval examples include the identity, reflection $t\mapsto 1-t$, and $t\mapsto 2t(\mathrm{mod}1)$. Invertibility is not required.

For an integer $K\ge 2$, multiplication by $K$ on the circle group preserves normalized Lebesgue measure. Each inverse image consists of $K$ branches, each scaling lengths by $1/K$.

The integer multiplication map on the circle is an ergodic transformation for Lebesgue measure. In the Fourier basis, an invariant function has coefficients $c_j=0$ when $K$ does not divide $j$, and $c_j=c_{j/K}$ otherwise. Every nonzero index eventually reduces to the first case, so only the constant coefficient remains.

A measure-preserving system is a measure space $(X,\mathcal{B},\mu )$ together with a measure-preserving transformation $T$. In ergodic theory the measure is commonly a probability measure.

A factor map $\pi$ is a measurable map satisfying $\pi \circ T=S\circ \pi$ and ${\pi }_{\ast }\mu =\nu$, from one measure-preserving system to another. The target system is the factor; its observable functions pull back to observable functions of the source.

A measurable measure-preserving map intertwining the transformations of two probability measure-preserving systems. The equalities may be imposed on invariant full-measure domains. It represents the target's observable information inside the source. The conditional measures of a factor describe the information remaining on each fiber.

An extension for which the relatively almost periodic observables are dense in global $L^2$. Density does not assert that every $L^2$ function already meets the uniform finite-net condition on fibers. Localization yields the positive-measure almost periodic indicator in a compact extension.

For a skew-product extension with finitely many labeled fibers, the orbit of a bounded observable on each fiber lies in a fixed bounded subset of a finite-dimensional vector space. A finite net there supplies global centers constant in the base coordinate. Bounded observables are dense in $L^2$, so this is a compact extension. Global almost periodicity can still fail because the base can be weakly mixing.

Approximate $1_A$ by relative almost periodic functions with summable squared global errors. Tonelli theorem makes their conditional errors tend to zero almost everywhere. Choose a positive base set on which the conditional mass of $A$ is bounded below, then use the Egorov theorem to make those errors uniform on a smaller positive set $C$. Localization of relative almost periodicity to base sets and conditional-norm closure prove $1_B$ is relatively almost periodic and $\mu (B)>0$.

For every positive $\epsilon$, require finitely many global $L^2$ centers $g_j$ such that the displayed inequality holds for every integer $n$ on almost every fiber of the factor map. A center can depend on the fiber and time; its finite list is fixed. This is weaker than almost periodic observable in global norm. For noninvertible systems use nonnegative times.

Multiplying a relatively almost periodic observable by a base-set indicator preserves relative almost periodicity: on each fiber an orbit value is either unchanged or zero, so add zero to the finite list of centers. Conditional norm covariance under a factor map also proves closure under convergence in the uniform conditional L2 norm.

Let $X$ be a finite set, let every singleton have positive probability, and let $T:X\to X$ preserve that probability measure. For every $x\in X$,so every $x$ has a preimage. Thus $T$ is a surjection, and every surjection from a finite set to itself is a bijection.

An invertible measure-preserving system has a measurable inverse modulo null sets and preserves its measure in both time directions. Its Koopman operator on $L^2$ is unitary.

The invariant sigma-algebra of $T$ consists, modulo null sets, of measurable sets $A$ satisfying $T^{-1}A=A$.

A measurable set $A$ satisfying $T^{-1}A=A$, or equality modulo null sets in the measure-theoretic convention. These sets form a $\sigma$-algebra, and an ergodic measure-preserving transformation gives each invariant set probability zero or one.

A measure-preserving transformation is ergodic when every set in its invariant sigma-algebra has measure zero or full measure. Equivalently, every integrable invariant function is almost everywhere constant.

A probability measure-preserving transformation is strongly mixing when $\mu (T^{-n}A\cap B)\to \mu (A)\mu (B)$ for every pair of measurable sets $A,B$. Unlike weak mixing, this is an ordinary limit along all sufficiently large times.

A probability measure-preserving transformation is strong mixing exactly when $\mu (T^{-n}A\cap A)\to \mu (A{)}^2$ for every measurable $A$. The converse follows by applying decay of autocorrelation implies weak convergence of an observable to $f=1_A$ and then testing against $1_B$. Diagonal polarization alone controls a sum of the two directed cross-correlations and does not suffice as a proof.

For a measure-preserving system on a standard probability space, an invariant probability measure is an integral of ergodic invariant probability measures. Its conditional measures given the invariant sigma-algebra are the ergodic components.

For rotation by $p/q$ with coprime integers and Haar probability on the circle, components are the equal measures on the $q$-point orbits. The quotient map is $x\mapsto qx(\mathrm{mod}1)$. Integration over the quotient recovers Haar measure, and the rotation permutes each orbit transitively, making its probability measure ergodic.

A conditional probability of an invariant probability measure given its invariant sigma-algebra. In a standard Borel probability system, almost every such probability is invariant and ergodic. Their integral recovers the original measure. The countable-test proof of ergodicity of conditional components explains why invariance alone is not the whole conclusion.

Apply the Birkhoff ergodic theorem to a countable generating algebra, and disintegrate the resulting common full-measure convergence set over the invariant sigma-algebra. The orbit limits on a fiber are its conditional masses. Applying the theorem to that invariant component shows its invariant projection is constant on all generating indicators, hence on all $L^2$ functions by density. This proves ergodicity without an uncountable intersection of null sets.