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For a bounded real function on a compact convex set, this envelope is the infimum of its continuous affine majorants. It is finite, concave and upper semicontinuous, and dominates the function. For continuous functions it describes the maximal integral among probability measures with a specified barycenter. Its homogeneity, subadditivity and affine-translation identities give the supporting measure lemma for affine upper envelopes.

An affine utility on probability measures functional satisfies $U_0(p\lambda +(1-p)\mu )=p{U}_0(\lambda )+(1-p)U_0(\mu )$ on mixtures of probability measures. Strict preferences are represented by comparing its real values, and indifference means equal values. This is cardinal utility on lotteries rather than merely an ordinal ordering of deterministic outcomes.

Andrey Kolmogorov formulated the modern measure-theoretic axioms of probability in his 1933 Grundbegriffe. A probability measure assigns total mass one and countably additive weights to events. This makes probability a mathematical structure that can be used across very different random experiments.^[ref]

An approximate design is a probability measure on the design region, with weight $w_i$ specifying a proportion of observations at $x_i$. For $N$ independent observations and common error variance ${\sigma }^2$, a linear regression with vector $f(x)$ has normalized information matrix of an experimental design $M(\xi )=\int f(x)f(x{)}^{T}d\xi (x)$; its least-squares estimator has covariance ${\sigma }^{2}M(\xi {)}^{-1}/N$ when the allocations realize these proportions. Exact designs restrict $N{w}_i$ to integers.

An arbitrage is an allowed self-financing portfolio with zero initial wealth, nonnegative terminal portfolio wealth almost surely under the physical probability measure $P$, and strictly positive terminal wealth with positive probability. In a finite model where every state has positive probability, nonnegativity means no loss in any state. Positive expected return alone does not establish arbitrage: losses may still occur.

For the binomial market example, suppose the European call option sells for 13 while its replicating portfolio costs $250/21$. Sell the call, buy its replicating portfolio, and lend the surplus $13-250/21=23/21$ in the bank account. The initial net cost is zero. At maturity the call obligation and portfolio cancel in either state, leaving $(21/20)(23/21)=23/20$. A price below $250/21$ permits the reversed construction, using short selling and borrowing. This is the law of one price made operational; the assumed trading permissions are essential.

Let $Z$ have the fair Bernoulli distribution and set $X_j=Z$ for every $j$. Each coordinate has the same fair probability distribution as in the independent model, but the coordinates are dependent: $\mathbb{P}(X_1=1,X_2=1)=1/2$, whereas the product of the marginal probabilities is $1/4$. Every future sigma-algebra equals $\sigma (Z)$, so the tail sigma-algebra also equals $\sigma (Z)$.

The tail event of infinitely many heads is now exactly $\{Z=1\}$, with probability $1/2$. For the finite-window events $B_{m,r}$ in infinitely many heads as a tail event,The sample space can even be the same full Cartesian product $\{0,1{\}}^{\mathbb{N}}$ in both models: the dependent probability measure assigns mass $1/2$ to each of the all-zero and all-one sequences. Then $A$ is the same literal tail event in both models. Only the probability measure and its independence properties change.

The barycenter of a probability measure on a compact convex set is the point satisfying the displayed affine integral identities. It exists by approximating the measure by finitely supported measures and using compactness of their finite convex combinations. It is unique because continuous affine functions separate points. In a Banach-space setting it agrees with the appropriate vector integral when that integral is defined.

For a probability measure whose identity map is strongly measurable with integrable norm, its barycenter is the Bochner integral of that map. It is characterized by $\phi (b(\mu ))=\int \phi (x)d\mu (x)$ for every continuous linear functional. For a measure on a weakly compact set in a separable Banach space, its barycenter belongs to the norm-closed convex hull of that set.

On a finite sample space $\Theta$, a basic probability assignment in Dempster-Shafer theory is a function on its power set satisfyingThe mass $m(B)$ is support committed to $B$ as a whole, without allocating it among the elements of $B$. Thus $m$ is a probability distribution over subsets, and generally is not an additive probability measure on $\Theta$.

For $\Theta =\{r,b\}$, let $m(\{r\})=0.6$ and $m(\Theta )=0.4$. The second mass expresses unresolved support between $r$ and $b$; it is not a claim that $b$ has probability $0.4$. This assignment induces belief function value $0.6$ and plausibility function value one for $\{r\}$. Ordinary additive probability is recovered when all nonzero masses are on singleton subsets.^[ref]

Suppose $\tilde{\mathbb{P}}$ is a probability measure with $d\tilde{\mathbb{P}}=Zd\mathbb{P}$, so $Z\ge 0$ and $\mathbb{E}Z=1$. For a sub-sigma-algebra $\mathcal{G}$, write $Z_{\mathcal{G}}=\mathbb{E}[Z\mid \mathcal{G}]$. If $X$ is integrable under $\tilde{\mathbb{P}}$, then on $\{Z_{\mathcal{G}}>0\}$,This is Bayes formula for conditional expectation. It changes the underlying probability measure as well as conditioning on the available information. The set where $Z_{\mathcal{G}}=0$ has zero $\tilde{\mathbb{P}}$-probability, so a version there can be chosen arbitrarily.

To verify the formula, let $Y$ denote the ratio. For $A\in \mathcal{G}$, measurability of $Y$ and the tower property give ${\int }_{A}Yd\tilde{\mathbb{P}}=\mathbb{E}[1_{A}ZY]=\mathbb{E}[1_A{Z}_{\mathcal{G}}Y]=\mathbb{E}[1_{A}ZX]={\int }_{A}Xd\tilde{\mathbb{P}}$. These are exactly the defining conditional-expectation identities. For a filtration, choosing $\mathcal{G}={\mathcal{F}}_t$ gives the usual time-indexed form. The denominator is the likelihood weight after reducing the full information to $\mathcal{G}$, not the original pointwise weight $Z$.

For $1<p\le q<\infty$ and $0\le \rho \le \sqrt{(p-1)/(q-1)}$, the noise operator on the Boolean hypercube satisfies $\Vert T_{\rho }f{\Vert }_q\le \Vert f{\Vert }_p$ for the uniform probability measure. In particular,Here $\hat{f}(S)$ are Fourier-Walsh transform coefficients. The endpoint $\rho =0$ follows by averaging. This hypercontractive inequality on the Boolean hypercube controls Fourier-Walsh transform coefficients of discrete derivatives of a Boolean function and is a principal ingredient in the Kahn-Kalai-Linial theorem.

For a measurable event $A$, applying Jensen inequality to $u\log u$ separately on $A$ and its complement shows that the Kullback-Leibler divergence between two probability measures is at least the divergence between the two event indicators. This reduces a general probability measure comparison to a binary one and is the key step in a proof of Pinsker's inequality.

For every $\epsilon >0$ there is a positive integer $K(\epsilon )$ such that every finite bipartite graph has an $\epsilon$-regular pair of partitions with at most $K(\epsilon )$ cells on each side.

To prove this, give $X\times Y$ the uniform probability measure, let $f$ be the indicator function of the edge set, and for partitions $\mathcal{P},\mathcal{Q}$ define their energy byThis conditional expectation is an orthogonal projection, so $0\le \mathcal{E}\le \Vert f{\Vert }_2^2\le 1$. If the partitions are not $\epsilon$-regular, choose witnesses $A_{ij}\subseteq X_i$ and $B_{ij}\subseteq Y_j$ in every irregular cell pair and refine each $X_i$ by all its $A_{ij}$ and each $Y_j$ by all its $B_{ij}$. The refined energy exceeds the old energy byOn $A_{ij}\times B_{ij}$ the absolute value function of the mean of the difference is greater than $\epsilon$, while this rectangle occupies at least an ${\epsilon }^2$ fraction of $X_i\times Y_j$. The Cauchy-Schwarz inequality therefore contributes more than ${\epsilon }^4|X_i||Y_j|/(|X||Y|)$ from that cell pair. Irregular pairs have total weight greater than $\epsilon$, so every refinement raises the energy by more than ${\epsilon }^5$. The process stops after at most $\lceil {\epsilon }^{-5}\rceil$ refinements. If the current cell counts are $r,s$, the construction gives at most $r{2}^s,s{2}^r$ new cells, so a finite iterated bound depending only on $\epsilon$ supplies $K(\epsilon )$.

Boole's inequality, also called the union bound, bounds the probability that at least one of several events occurs by adding their individual probabilities. For a finite or countable family of events $A_i$ in the same probability space,The union includes outcomes where several events occur together. There is no independence assumption: adding the individual probabilities counts such outcomes repeatedly, which can only make the sum larger. The cap at one uses the total mass of a probability measure. For infinitely many events, the sum is a nonnegative series, possibly infinite; an infinite sum gives only the bound one. This is countable subadditivity of a measure specialized to a probability measure. See Probability: Theory and Examples (fifth edition) by Rick Durrett (Section 1.1, Theorem 1.1.1(ii), printed p. 2).

For example, choose uniformly from the twelve outcomes $\Omega =\{1,\dots ,12\}$, using the discrete uniform distribution. Put $A=\{1,2,3,4\}$. If $B=\{5,6,7,8\}$, these are disjoint events and their union has probability $8/12=2/3$, exactly the union bound. If $B=\{3,4,5,6\}$, the union has only six outcomes, so its exact probability is $1/2$, although both individual probabilities are still $1/3$ and the bound is still $2/3$. If $B=A$, the exact probability falls to $1/3$ with the same bound. These examples isolate overlap in the union bound by keeping the individual probabilities fixed.

The union bound is useful when each event is easy to control but their joint behavior is difficult to determine. A short proof uses indicator functions: at every outcome,Taking expected values and using linearity of expectation proves the finite bound.^[ref] For a countable family, the finite unions increase to the full union; continuity of probability passes the finite inequalities to the limit. The closure of a sigma-algebra under countable unions ensures that this limiting union is an event. For the continuity of probability step, see Probability: Theory and Examples (fifth edition) by Rick Durrett (Section 1.1, Theorem 1.1.1(iii), printed p. 2).

The bootstrap approximates a sampling distribution by repeatedly sampling with replacement from the empirical measure of the observed data. In the ordinary nonparametric version, observations are assumed independent and identically distributed; a bootstrap sample has the same size as the original statistical sample. A statistic, such as a sample mean or sample median, is recomputed on each resample. Its variation across resamples estimates how the original statistic would vary across repeated sampling from the unknown probability distribution.^[ref]

The key distinction is between the observed statistical sample and its still-unknown source probability distribution. Condition on the observed values $x_1,\dots ,x_n$ and form the empirical measureHere ${\delta }_{x_i}$ is a Dirac measure. Draw independent indices $I_1,\dots ,I_n$ from the uniform distribution on a finite set $\{1,\dots ,n\}$ and set $X_j^{\ast }=x_{I_j}$. Each observed slot has probability $1/n$ on every draw. If equal values occupy several slots, their total probability is the sum of those slot probabilities; distinct observed values need not be equally likely. A star denotes this conditional resampling experiment. There are $n^n$ equally likely ordered index sequences, although bootstrap count vectors and statistics can combine many such sequences.

A complicated sampling distribution need not stop statistical inference when recomputing the statistic is easy. The bootstrap reuses that calculation on simulated datasets. The plug-in estimator principle replaces the unknown probability measure by ${\hat{P}}_n$ in the whole repeated-sampling experiment. Whether that substitution approximates the target sampling distribution needs an appropriate bootstrap consistency theorem for the sample mean or a result for the particular statistic. A larger number of resamples reduces Monte Carlo error; it cannot repair the bootstrap failure for a uniform endpoint, an unrepresentative statistical sample or ignored statistical dependence.^[ref]

The bounded-Lipschitz metric tests probability measures against a uniformly bounded Lipschitz class. It metrizes weak convergence of probability measures on separable metric spaces. With a complete compatible base metric, it is complete on the probability measures of a Polish space. Replacing the sum norm by the maximum norm changes the metric only by uniform constant factors.

A bracket $[\ell ,u]$ contains every measurable function $h$ satisfying $\ell \le h\le u$ pointwise. For a Probability measure $P$, its $L^1(P)$ width is $P(u-\ell )$, with integrable endpoints. Finite bracketing at every positive width is a sufficient complexity condition for a uniform law of large numbers. Pointwise inequalities, or inequalities outside a single common null set, allow empirical inequalities to hold simultaneously throughout the class.

For probability measures on ${\mathbb{R}}^d$ with finite second moments and a source satisfying absolute continuity of measures with respect to Lebesgue measure, the quadratic Kantorovich optimal transport problem has a unique optimal transport plan. It is induced by the gradient of a convex function, which also uniquely solves the Monge optimal transport problem up to a source-null set.

Cantor space is the binary sequence space with the product topology of discrete two-point factors. Here the Bernoulli-space synonym refers to this topology, without selecting a particular probability measure. The metric $d(x,y)={\sum }_{j\ge 1}{2}^{-j}|x_j-y_j|$ induces the topology. A diagonal subsequence argument gives compactness. Fixing a finite prefix gives a clopen cylinder set. The map $x\mapsto {\sum }_{j}2x_j{3}^{-j}$ is a homeomorphism to the usual Cantor set.

The centered Cauchy densitiesform a convolution semigroup: $K_a\ast K_b=K_{a+b}$. With Fourier convention ${\hat{K}}_a(t)=\int e^{itx}{K}_a(x)dx$, the characteristic function is $e^{-a|t|}$. Multiplying characteristic functions proves the semigroup identity. Equivalently, the sum of independent centered Cauchy random variables has scale equal to the sum of their scales; for example, scales two and three combine to scale five.

This family permits an exact deconvolution calculation for a particular right-hand side. If $a>b>0$, the integral equationhas probability-density solution $f=K_{a-b}$. If $a=b$, the probability-measure solution is the Dirac measure at zero rather than an ordinary density. If $a<b$, a probability-measure solution would have characteristic function $e^{(b-a)|t|}$, which exceeds one in modulus for nonzero $t$ and is impossible.

Subtracting scales works because both sides belong to this convolution family. For a general right-hand side, Fourier deconvolution instead multiplies its transform by $e^{b|t|}$, potentially amplifying high-frequency errors. The semigroup identity does not make arbitrary Cauchy-kernel integral equations stable or solvable by a scale subtraction.