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For a function $f:X\to Y$ and a subset $U\subseteq Y$, its preimage isIt is a subset of the original domain of a function. The notation does not require an inverse function: the preimage of a singleton set can have no members, one member, or many members. In the finite example, $f^{-1}(\{0\})=\{a,b\}$, $f^{-1}(\{1\})=\{c\}$ and $f^{-1}(\{3\})=\varnothing$. The preimage of a singleton set is also its fiber of a function.

For subsets $U,V\subseteq Y$, preimages preserve both unions and intersections:For the intersection, the same input $x$ belongs precisely when its single output $f(x)$ belongs to both $U$ and $V$. There is no opportunity to choose different witnesses, which is what made the image counterexample fail. The identities also hold for arbitrary families of output subsets.

In the finite example, choose $U=\{0,1\}$ and $V=\{0,2\}$. ThenThe intersection of the two preimages is exactly the same set $\{a,b\}$. Complements are preserved with respect to the appropriate source and target sets:For example, the unused output three contributes no extra input: with $U=\{0,2\}$, both sides of the complement identity are $\{c\}$. See Lebl, Proposition 0.3.15: preimages of unions, intersections and complements.

This makes preimages useful for transporting conditions. The solution set of $g(x)=0$ is $g^{-1}(\{0\})$, and a zero set is a preimage of zero. Requiring two output conditions takes the intersection of their preimages. In topology, a continuous function pulls every open set back to an open set; a measurable function pulls the specified measurable sets in its codomain back to measurable sets in its domain of a function. These definitions use preimages because their set operations remain consistent even when the function is noninjective.

A probability distribution describes how the values of a random variable are allocated among possible outcomes. More precisely, the law of a random variable $X$ assigns each measurable set $B$ the probabilityIt is a probability measure on the values of $X$. The underlying experiment can be more elaborate than the output: different experiments may produce the same probability distribution. The law of a random variable retains the probabilities needed to answer questions about that output. ^[ref].

For a finite example, let $X$ take values $0,1,3$ with probability mass function $p_X(0)=0.2$, $p_X(1)=0.5$, $p_X(3)=0.3$. Its probabilities sum to one. The event $0<X\le 1$ has probability $0.5$, whereas $0\le X\le 1$ has probability $0.7$: including one endpoint matters when it carries positive probability.

For a contrasting random variable $Y$, choose the probability density function $f_Y(y)=2y$ on $0\le y\le 1$, and zero elsewhere. It integrates to one. Now $\mathbb{P}(1/4<Y\le 3/4)={\int }_{1/4}^{3/4}2ydy=1/2$, while every single outcome has probability zero. The probability density function can have height $f_Y(0.8)=1.6$: probabilities come from integrals over outcome ranges.

The cumulative distribution function answers all threshold questions for a real random variable, regardless of whether there are jumps or a probability density function. A quantile function reverses the direction of the question: given a cumulative probability level, find a threshold. This connects descriptive summaries to inverse transform sampling, which constructs a chosen probability distribution from a uniform distribution.

An outer measure is regular in this sense when every set admits a measurable hull of equal outer measure. This is different from merely specifying inner or outer approximation of already measurable sets by compact or open sets. Lebesgue outer measure has a hull that is a Borel set: for finite outer measure choose open covers of total volumes within $1/j$ of its value and intersect their unions. Monotonicity gives equality of the hull's outer measure with that of the original set. For infinite outer measure the whole Euclidean space is a hull. This regularity is useful for turning numerical equality of inner and outer sizes into measurability.

For a measure space $(\Omega ,\mathcal{F},\mu )$ and $E\in \mathcal{F}$, the restriction of a measure to a measurable subset is the measure on $(E,\mathcal{F}{|}_E)$ given by the displayed formula. The trace sigma-algebra consists of the measurable sets contained in $E$. The empty-set axiom and countable additivity are inherited directly from $\mu$. An associated measure on the original sigma-algebra is $\nu (A)=\mu (A\cap E)$ for $A\in \mathcal{F}$; the two constructions have different underlying measurable spaces.

A set function assigns a value to each set in a specified family of sets. In finite optimization the usual form is $f:\mathcal{P}(V)\to \mathbb{R}$, a real-valued function on the power set of a ground set $V$.

For a finite set $V$, the rule $f(S)=|S|$ is a set function counting selected elements; $f(S)={\sum }_{v\in S}w(v)$ instead totals their weights. A measure is also a set function, but its domain of a function is a specified collection of measurable sets, and its values can include infinity. The intended family and codomain are therefore part of the definition.

A measurable set $E$ has finite perimeter in an open domain $\Omega$ when the distributional derivative of ${\chi }_E$ is a finite vector Radon measure there:The indicator is always locally integrable. Equivalently it belongs locally to BV space and has finite total derivative variation on the whole domain. If $|E\cap \Omega |<\infty$, it also belongs to the global BV space with its L1 norm. Finite perimeter itself does not require that finite-volume condition.

For example, the half-line $(0,\infty )$ has perimeter one in $\mathbb{R}$, from its single endpoint Dirac measure, despite infinite length. The entire real line has perimeter zero, although its indicator is not in $L^1(\mathbb{R})$. On a bounded domain, finite volume is automatic, so the simpler condition ${\chi }_E\in BV(\Omega )$ is equivalent.

Relative perimeter counts the interface inside $\Omega$ and excludes boundary outside that open domain. In a Caccioppoli partition, each shared interior interface appears in the perimeters of both adjacent regions; a segmentation cost must account for that double counting.

A signed measure is a countably additive set function on a sigma-algebra that permits negative values. In the extended-real convention it takes values in $[-\infty ,\infty ]$, has value zero on the empty set, and does not take both $+\infty$ and $-\infty$, so its additive sums are defined. The important finite case is real-valued on every measurable set.^[ref]

For distinct points $a,b$, the difference of Dirac measures $\nu ={\delta }_a-2{\delta }_b$ is a finite signed measure. It has $\nu (\{a\})=1$, $\nu (\{b\})=-2$ and $\nu (\{a,b\})=-1$. Its variation measure is $|\nu |={\delta }_a+2{\delta }_b$, whose total mass is $3$, rather than the absolute value $1$ of its signed total mass. The total variation norm of a measure records both contributions before cancellation.

A Hahn decomposition separates a signed measure into a region on which all measurable subsets have nonnegative mass and a region on which all measurable subsets have nonpositive mass. It is the setwise counterpart of splitting an integrable function into its positive part and negative part.

Steiner symmetrization about a line replaces every perpendicular slice of a measurable set by a centered interval of the same one-dimensional Lebesgue measure. For a bounded Borel set $E\subset {\mathbb{R}}^2$, take the horizontal axis as the symmetry line and putSlice endpoints do not affect area. A slice can consist of several disjoint intervals: its total length determines the replacement.^[ref]

For a convex set bounded by graphs $u_{-}(x)$ and $u_{+}(x)$, the new bounds are $\pm (u_{+}-u_{-})/2$. A disk centered on the symmetry axis stays unchanged. Fubini's theorem proves $\mathrm{area}(E^{\ast })=\int {\ell }_E(x)dx=\mathrm{area}(E)$. Preserved area together with perimeter decrease under Steiner symmetrization makes the operation useful for the isoperimetric inequality.

On a real interval, a step function is constant on the open cells of a finite partition of an interval, with endpoint values specified separately. Equivalently, up to those finitely many point values it is a finite linear combination of interval indicator functions. For example, a function equal to two on $[0,1/4)$, one on $[1/4,3/4)$ and zero on $[3/4,1]$ is a step function. Its integral is the sum of rectangle heights times widths.

A simple function is more general in measure theory: its finitely many value sets can be arbitrary measurable sets. The Dirichlet function $1_{\mathbb{Q}}$ on $[0,1]$ is simple but is not an interval step function, because every nonempty interval contains both rational and irrational points. Merely being constant on a finite or countable measurable partition therefore does not characterize the usual interval-based step functions.

Countably many jumps can be considered under explicitly stated extensions of the term. The finite-partition convention is the one used for elementary Riemann approximations; the measurable finite-range convention belongs to simple functions in Lebesgue integration.

The union of independent Poisson point processes with diffuse intensity measures ${\mu }_1,\dots ,{\mu }_m$ is a Poisson point process with intensity measureOn every finite collection of disjoint measurable sets, this follows by adding independent Poisson counts and checking that the resulting counts remain independent.

Total variation distance asks how much two probability distributions can disagree about the same event. For probability measures $P,Q$ on one measurable space $(S,\mathcal{F})$, its definition isThe supremum ranges over all measurable sets, so $d_{\mathrm{TV}}(P,Q)\le \epsilon$ guarantees that every yes-or-no prediction has probability error at most $\epsilon$. This metric takes values between zero and one; it is zero exactly when the two probability measures agree. This makes it useful for comparing a sampling algorithm's output with its target probability distribution, and for defining the mixing time of a Markov chain.

On a finite set, add the excess probability mass wherever $P$ exceeds $Q$. That excess is also half the sum of all absolute differences:The last expression reveals the common probability mass that the two probability distributions can share.^[ref] The coupling characterization of total variation distance turns this overlap into an actual joint construction; minimum sum of errors in a simple hypothesis test gives it a statistical interpretation. One quantity connects probability bookkeeping, comparison of sampling methods, and the limits of a statistical test.

For a measurable space $(\Omega ,\mathcal{F})$ and any subset $E\subseteq \Omega$, the trace sigma-algebra is the displayed family of subsets of $E$. It contains $E=E\cap \Omega$, is closed under relative complements since $E\setminus (E\cap A)=E\cap (\Omega \setminus A)$, and is closed under countable unions. If $E\in \mathcal{F}$, it equals $\{B\in \mathcal{F}:B\subseteq E\}$. Without that measurability assumption, the traces need not be measurable sets in the original sigma-algebra.

A truncated normal distribution is a normal distribution conditioned on belonging to a retained measurable set $A$ of positive probability. If $X\sim N(\mu ,{\sigma }^2)$, $\sigma >0$, and $Z=\mathbb{P}(X\in A)>0$, its density iswhere $\varphi$ is the standard normal density. The indicator is essential: the density is zero outside $A$. For an interval $A=(a,b)$, possibly with infinite endpoints,The same expression holds for any endpoint inclusion convention because the original normal distribution has no atoms.

For a standard normal variable restricted to positive values, $Z=1/2$ and the density is $2\varphi (x)$ for $x>0$. Its mean is $\sqrt{2/\pi }$: integrate $2x\varphi (x)$ using ${\varphi }^{\prime }(x)=-x\varphi (x)$. Symmetry gives second moment one, hence variance $1-2/\pi$. Truncation therefore changes the mean and variance; they are not generally the original parameters $\mu ,{\sigma }^2$.

sign-truncated normal sampling uses inverse transform sampling to generate this conditional law directly. Rejection from the original normal is also exact but may be slow when the retained set has very small probability.

A finite-dimensional vector measure is a countably additive map $\mu :\mathcal{F}\to {\mathbb{R}}^d$ on a sigma-algebra. Equivalently, each component is a finite signed measure. For pairwise disjoint measurable sets $A_n$, countable additivity means $\mu ({\bigcup }_n{A}_n)={\sum }_n\mu (A_n)$ with convergence in the vector norm.

Its variation measure records size without cancellation:The supremum ranges over finite measurable partitions of $A$. For $\mu (A)=(\lambda (A),2\lambda (A))$ on $[0,1]$, the Euclidean norm gives $|\mu |(A)=\sqrt{5}\lambda (A)$, rather than the sum $3\lambda (A)$ of component variations. The chosen vector norm matters.

The polar decomposition of a vector measure writes $\mu =\sigma |\mu |$ with unit direction $\sigma$ almost everywhere for $|\mu |$. It separates local direction from variation magnitude. Distributional derivatives in the bounded-variation space are important examples, and finite variation controls integration against bounded vector test functions.

The void probability functional assigns to a measurable set the probability that it contains no atom of a random point measure. For a Poisson random measure of intensity $\mu$, it is $e^{-\mu (A)}$. The exponential void probabilities characterize a Poisson random measure result uses simplicity; doubling every atom of a Poisson measure preserves void probabilities but changes count distributions.