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For pairwise disjoint measurable sets $E_n$, countable additivity means

The infinite series is the limit of its nonnegative partial sums. This controls an entire countable union, not merely each finite stage. If $q_1,q_2,\dots$ enumerate the rational numbers in $[0,1]$, their finite collections have Lebesgue measure zero, and countable additivity gives their whole union Lebesgue measure zero. Their being a dense set does not change this calculation.

The axiom makes no assertion about an arbitrary uncountable set of disjoint pieces. In particular, $[0,1]$ is the union of its singletons, each of Lebesgue measure zero, yet its Lebesgue measure is one. Extending additivity to all such unions would contradict interval length.

A left Haar measure on a locally compact group that is a Hausdorff space is a nonzero regular Borel measure, finite on compact sets, satisfying $\mu (gA)=\mu (A)$ for every Borel set $A$. Such a measure exists and is unique up to a positive scalar. Right Haar measure instead satisfies $\mu (Ag)=\mu (A)$; the two need not coincide. On a compact group, normalization gives total mass one and invariance on both sides. See Tornier, Haar Measures, Definition 2.1 and Theorem 2.2.

For a finite group, normalized Haar measure is uniform counting; for the circle group, it is normalized angle integration on the circle. These examples turn symmetry into a rule for averaging.

If a measure $\nu$ has density $w=d\nu /d\mu$ with respect to $\mu$, thenThis identity is the basis of likelihood ratios and importance sampling.

For a measurable map $f:X\to Y$, the pushforward of $\mu$ is the measureIt is the distribution induced on $Y$ by mapping points distributed according to $\mu$ through $f$.

A measure space is a set $X$, a sigma-algebra $\mathcal{M}$ of its subsets, and a measure $\mu$ defined on $\mathcal{M}$. Specifying only $X$ leaves both the measurable questions and their weights undetermined.

For a concrete probability space, take $X=\{a,b,c\}$, $\mathcal{M}$ its power set, and weights $1/2,1/3,1/6$. Define $\mu (A)$ by adding the weights of the elements in $A$. ThenA probability measure is exactly a measure normalized by $\mu (X)=1$, usually written $\mathbb{P}(\Omega )=1$. An indicator function has expected value equal to its event's probability. For a different continuous example, $[0,1]$ with the restricted Lebesgue measure is a probability space; the event $[0,1/4]$ has probability $1/4$.

If a harmonic function on a connected domain attains an interior maximum or minimum, it is constant. The mean value property forces equality throughout every sufficiently small ball around an interior extremum, and connectedness propagates the equality.

This is the strong harmonic instance of the maximum principle.

If a continuous function has the spherical mean value property at every point for some sequence of radii decreasing to zero, then it is harmonic. Comparison on each relatively compact ball with the harmonic function having the same boundary data proves the result by propagating any hypothetical interior extremum along one of the admissible spheres.

For $X\sim \mathrm{Bin}(n,\theta )$ andthe mean squared error isWhen ${\theta }_0=1/2$, its maximal risk is at $\theta =1/2$ or at an endpoint of $[0,1]$.

Every estimator with finite second moment satisfies

Resonance capture occurs when slow orbital migration carries a body into a resonance and the resonant argument begins to librate, so resonant torque can balance the migration torque.

A resonant argument is a slow angular combination whose libration diagnoses resonance. For an exterior $j:(j-1)$ eccentric resonance it can be $\varphi =j\lambda -(j-1){\lambda }_p-\varpi$.

A resonant argument is an integer combination of orbital angles whose slow variation can diagnose mean-motion resonance. Its libration, rather than circulation through every angle, characterizes trapping in the resonance.

A first-order mean-motion resonance has $|j-k|=1$. For small eccentricity its leading eccentric resonant term is linear in eccentricity.

Let $f$ be bounded and holomorphic on a half-plane and continuous on its closure. A bound $|f|\le M$ on the boundary line propagates throughout the half-plane. Apply the bounded-domain principle to $f(z)z^{-1/n}$ on expanding truncated half-discs, then let $n\to \infty$.

Writing $a={\Sigma }_{11}$, $c={\Sigma }_{12}$, and $b={\Sigma }_{22}$, the vector $({\hat{\Sigma }}_{11},{\hat{\Sigma }}_{12},{\hat{\Sigma }}_{22})$ has asymptotic covariance

Local class field theory describes the abelian extensions of a local field $K$ through its multiplicative group $K^{\times }$.

The local Kronecker-Weber theorem states that every finite abelian extension of ${\mathbb{Q}}_p$ is contained in an extension obtained by adjoining roots of unity.

The cyclotomic extension ${\mathbb{Q}}_p({\zeta }_{p^n})/{\mathbb{Q}}_p$ is generated by a primitive $p^n$th root of unity. It is totally ramified and has Galois group $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$.

For a non-Archimedean local field $K$, the Weil group is the inverse image of the infinite cyclic subgroup generated by Frobenius under $\mathrm{Gal}(\overline{K}/K)\to \mathrm{Gal}(\overline{k}/k)\cong \hat{\mathbb{Z}}$. It is dense in the absolute Galois group with its profinite topology.

If one edge leaving the source has capacity $x$, every cut capacity is affine in $x$ with coefficient zero or one. In the 2024 Cambridge Part IB example, cuts of capacities $x+5$ and $14$ are both sharp, giving

If a symmetric invertible payoff matrix satisfies $A^{-1}e\ge 0$, both players may useand the game value is $(e^T{A}^{-1}e{)}^{-1}$.