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Every nonnegative measurable function is the pointwise increasing limit of nonnegative simple functions, and its integral is the supremum of their integrals.

If two measures agree on a generating $\pi$-system and the space is a countable union of members having finite common measure, then they agree on the generated $\sigma$-algebra.

Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be sigma-finite measure spaces, and let $f$ be measurable for the product sigma-algebra. Ifthen its sections are integrable for almost every fixed coordinate, the resulting section integrals are integrable, andValues on the exceptional null sets can be assigned arbitrarily. The complex-valued version follows by real and imaginary parts.^[ref]

A useful order of reasoning is to apply Tonelli theorem to $|f|$ first. Once that nonnegative double integral is finite, Fubini justifies either signed order. For $f(x,y)=xy$ on $[0,1{]}^2$, absolute integrability is immediate and either order gives $1/4$. Merely obtaining two finite iterated signed integrals does not prove joint integrability; finite iterated integrals without joint integrability gives an explicit counterexample. The absolute-value hypothesis prevents hidden cancellation of infinite positive and negative contributions.

A sigma-algebra on a set $\Omega$ is a collection $\mathcal{F}$ of subsets such that $\Omega \in \mathcal{F}$, $A\in \mathcal{F}$ implies $\Omega \setminus A\in \mathcal{F}$, and $A_n\in \mathcal{F}$ for all $n$ implies ${\bigcup }_n{A}_n\in \mathcal{F}$. Thus it contains the empty set and is closed under countable intersections as well. This is precisely the closure needed to apply countable additivity without leaving the domain of a measure.

For $\Omega =\{a,b,c\}$,is a sigma-algebra. It describes observations that distinguish $a$ from the pair $\{b,c\}$, but cannot distinguish $b$ from $c$. The power set provides a finer sigma-algebra with all eight subsets. Both are valid; choosing a sigma-algebra records which questions the model can answer.

On the real numbers, the Borel sigma-algebra contains all open sets and everything generated from them by these operations. The Lebesgue measurable sets form a larger sigma-algebra, including all subsets of Borel null sets. Closure under countable unions accommodates constructions involving an entire sequence of observations or approximations.

The indicator function of a set $E$ is $1_E(x)=1$ for $x\in E$ and $0$ otherwise.

On a measurable space $(X,\mathcal{F})$, a measure is a function $\mu :\mathcal{F}\to [0,\infty ]$ with $\mu (\varnothing )=0$ and countable additivity: for pairwise disjoint measurable sets $A_n$,Infinite size is permitted. Nonnegativity implies that if measurable sets satisfy $A\subseteq B$, then $\mu (A)\le \mu (B)$: decompose $B$ into $A$ and $B\setminus A$. For overlapping measurable sets, use countable subadditivity rather than adding their full sizes twice.

For example, counting measure gives $\mu (\{a,b,c\})=3$, while a Dirac measure ${\delta }_a$ gives that same set size $1$ if it contains $a$. Under Lebesgue measure, $[0,1/4]\cup [3/4,1]$ has size $1/4+1/4=1/2$. These are all measures, but they answer different questions: how many objects, whether a distinguished object is present, or how much length is occupied.

Ergodic theory studies the long-time statistical behavior of measure-preserving systems. A point follows a deterministic orbit $x,Tx,T^{2}x,\dots$, while an invariant measure describes how an ensemble of points is distributed. For an integrable observable $f$ on a probability space, the basic question is what the time averagesreveal about the spatial average $\int fd\mu$. The Birkhoff ergodic theorem gives almost-everywhere convergence; ergodicity makes the limit the same spatial average for almost every starting point.^[ref]

For example, let $X=\{0,1\}$ with equal probabilities and $T$ exchange the two points. For $f(x)=x$, every orbit average tends to $1/2$, and the system is ergodic. Yet it is not strong mixing: with $A=\{0\}$, $\mu (T^{-n}A\cap A)$ alternates between $1/2$ and zero rather than tending to $\mu (A{)}^2=1/4$. Thus agreement of long-time averages does not require loss of every time correlation.

The Koopman operator translates these questions into linear operator language, and entropy rate of a measurable partition quantifies how much information an orbit reveals per step. Poincaré recurrence theorem concerns returns of individual trajectories, while Furstenberg multiple recurrence theorem connects simultaneous returns to arithmetic progressions. These are distinct ways of extracting structure from a deterministic evolution.

The conditional expectation is the average of a random variable that can be predicted using specified information. It is itself a random variable: the average appropriate to one observed group may differ from that appropriate to another group. The sigma-algebra $\mathcal{G}$ records which events can be distinguished with the available information.

For a real integrable random variable $X$ on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ and a sub-sigma-algebra $\mathcal{G}$, the conditional expectation $m=\mathbb{E}[X\mid \mathcal{G}]$ is the almost surely unique integrable random variable that is measurable with respect to $\mathcal{G}$ and satisfiesIts average therefore agrees with that of $X$ on every distinguishable event. The integral characterization handles discrete observations, continuous observations, and information that is not given by a single numerical value.^[ref]

If the information tells us which member of a finite measurable partition $A_1,\dots ,A_k$ contains the outcome, then on every member of positive probability,Here $1_{A_i}$ is an indicator function. Zero-probability members do not determine a unique value, which is why uniqueness is only almost surely. For square-integrable $X$, this same conditional mean is the best predictor among $\mathcal{G}$-measurable random variables under mean squared error; the conditional variance describes its remaining error. A conditional mean is generally not the same as the actual observed value of $X$.^[ref]

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

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

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.

Let $A\subseteq {\mathbb{R}}^d$ be measurable and let $\mu$ be a locally finite positive Borel measure. At a point in the support of a measure of $\mu$, all sufficiently small balls have finite positive mass. Its relative density iswhen the limit exists. The ratio lies in $[0,1]$. The support condition prevents zero denominators; at a point outside the support, sufficiently small balls have no mass and this ratio is undefined without an extra convention.

For Lebesgue measure on the real line and $A=[0,1]$, the density is one at interior points, zero at exterior points and one-half at either endpoint. For $\mu ={\delta }_0$, the density at its support point is $1_A(0)$. This is a fraction of one measure's local mass. It differs from the spherical derivative of a measure, which compares the mass of a measure with Euclidean volume. The Lebesgue density theorem gives the almost-everywhere zero-or-one statement when the reference is Lebesgue measure.

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

Projection onto the mean-variance portfolio $V^{-1}b$ leaves residual covariance $V-b{b}^T/(b^T{V}^{-1}b)$.

A portfolio dominates another when it has no smaller mean and no larger variance, with at least one strict improvement.

Let $\xi \sim N(b,\Sigma )$, where $b\ne 0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+{\theta }^T\xi$, every unique optimal portfolio has the formIndeed, the component orthogonal to ${\Sigma }^{-1}b$ in the $\Sigma$ inner product contributes independent mean-zero Gaussian risk without changing the mean. Removing it cannot reduce expected concave utility. A negative coefficient is dominated by the corresponding positive coefficient, which has the same variance and a larger mean.

Without a risk-free intercept or constraints, every undominated linear portfolio lies on the nonnegative ray through $V^{-1}b$.

A linear anisotropic dielectric has a direction-dependent linear relation between the electric displacement field and electric field:The permittivity tensor need not be proportional to the identity matrix, so the two fields need not be parallel. A corresponding linear magnetic response is $B_i={\mu }_{ij}{H}_j$, where $\mathbf{\mu }$ is the magnetic permeability tensor.

For an instantaneous, nondispersive, lossless model with real time-independent symmetric matrices, the field energy density isDifferentiation gives ${\partial }_{t}u=\mathbf{E}\cdot {\partial }_t\mathbf{D}+\mathbf{H}\cdot {\partial }_t\mathbf{B}$, the storage term in Poynting theorem in a linear anisotropic medium. Positivity of this quadratic energy density requires positive-definite response tensors in this model. Dispersive or absorptive material response requires a more general energy accounting.

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the maximum modulus principle places every maximum on the boundary.

In a regular $p$-parameter model with one-observation Fisher information $I({\theta }_0)$,