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Paths with common endpoints are internally vertex-disjoint when they share no other vertex.

An $x$-$y$ vertex separator is a set of vertices, excluding $x$ and $y$, whose deletion leaves no path from $x$ to $y$.

On a finite measure space, $L^q\subseteq L^p$ for $0<p<q\le \infty$. For $1\le p<q<\infty$, Hölder's inequality giveswith the same formula interpreted as $\Vert f{\Vert }_p\le \mu (X{)}^{1/p}\Vert f{\Vert }_{\infty }$ when $q=\infty$.

An increasing right-continuous function $F:\mathbb{R}\to \mathbb{R}$ determines a measure byThe Radon-Nikodym theorem and Lebesgue differentiation theorem identify the density of the absolutely continuous part of this measure with $F^{\prime }$ almost everywhere.

A measure $\nu$ is absolutely continuous with respect to $\mu$, written $\nu \ll \mu$, when every $\mu$-null measurable set is also $\nu$-null. This is distinct from the regularity property of functions called absolute continuity.

A measure $\nu$ is absolutely continuous with respect to $\mu$ when every $\mu$-null set is also $\nu$-null.

Tonelli’s theorem permits interchange of integrals for a nonnegative measurable function, even before finiteness is known.

On a finite measure space, if measurable functions are uniformly bounded and converge pointwise, then their integrals converge. This is the dominated convergence theorem with a constant dominating function.

If $f_n\to f$ almost everywhere and $|f_n|\le g$ for one integrable function $g$, then $f$ is integrable and $\int |f_n-f|\to 0$.

If nonnegative measurable functions satisfy $f_n\uparrow f$ almost everywhere, then $\int f_n\uparrow \int f$, allowing the value infinity.

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.

Fubini's theorem permits the order of integration of an absolutely integrable function on a product measure space to be exchanged.

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 behaviour of measure-preserving transformations.

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 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.