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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.

A change of measure replaces a positive measure $\mu$ by a positive measure $\nu$ described by a nonnegative Radon-Nikodym derivative $w=d\nu /d\mu$. For a nonnegative measurable function $g$, or for a function $g$ that is integrable with respect to $\nu$,Thus the integration weight changes while the underlying measurable space stays fixed. A density representation requires absolute continuity of measures $\nu \ll \mu$; the Radon-Nikodym theorem provides such a density under its hypotheses.

On $X=\{0,1\}$, let $\mu$ assign mass $1/2$ to each point and let $\nu$ assign masses $1/4,3/4$. Then $w(0)=1/2$ and $w(1)=3/2$, soFor probability measures, the identity becomes ${\mathbb{E}}_{\nu }[g]={\mathbb{E}}_{\mu }[wg]$ and ${\mathbb{E}}_{\mu }[w]=1$. It underlies likelihood ratios and importance sampling; strict positivity $\mu$-almost everywhere makes the two probability measures equivalent.

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 an integer linear combination of orbital angles that evolves slowly near an orbital resonance. For an outer body near an exterior $j:(j-1)$ mean-motion resonance with a nearly circular inner planet, one example isHere $\lambda$ and ${\lambda }_p$ are mean longitudes, $\varpi$ is the outer body's longitude of periapsis, and the corresponding mean motions are $n$ and $n_p$. Thus $\dot{\varphi }=jn-(j-1)n_p-\dot{\varpi }$ is small near the resonance, including the apsidal precession correction.

In resonant-argument libration, an unwrapped version of $\varphi$ oscillates around a centre rather than repeatedly running through every angle. Slow variation at one instant alone does not establish trapping. At equal mean longitudes, the example gives $\varphi ={\lambda }_p-\varpi$: its bounded range constrains where mean conjunctions occur relative to periapsis. This connects the angular diagnostic with resonant conjunction geometry and possible protection from close encounters.

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 the right half-plane, continuous on its closure, and satisfy $|f(iy)|\le M$ for every real $y$. Then $|f(z)|\le M$ throughout the half-plane. Rotations and translations give the analogous statement for any half-plane.

For $\epsilon >0$, set $g_{\epsilon }(z)=f(z)/(1+\epsilon z)$. Its denominator has no zero in the closed right half-plane and has modulus at least one on the imaginary boundary. If $|f|\le K$, then on the radius-$R$ semicircle, $|g_{\epsilon }|\le K/(\epsilon R-1)$. The maximum modulus principle on a bounded domain therefore gives $|g_{\epsilon }|\le M$ after letting $R\to \infty$; equivalently first use $M+\eta$ and then $\eta \downarrow 0$ when $M=0$. Finally let $\epsilon \downarrow 0$.

Boundedness in the half-plane cannot be replaced by a boundary bound alone: $e^z$ has modulus one on the imaginary axis and is unbounded to its right. Growth-controlled extensions require a Phragmén–Lindelöf principle.

Let $Z_i=(X_i,Y_i)$ be independent, identically distributed centered bivariate normal vectors with covariance matrix $\Sigma =\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, and let $\hat{\Sigma }=n^{-1}{\sum }_i{Z}_i{Z}_i^{\mathsf{T}}$. ThenThe asymptotic covariance $V$ belongs to the $\sqrt{n}$-scaled error; the unscaled estimator has covariance $V/n$, in fact exactly for every $n$. The distinction prevents a missing-sample-size error.

To derive it, apply the multivariate central limit theorem to the independent vectors $(X_i^2,X_i{Y}_i,Y_i^2)$. The Isserlis theorem gives $\mathbb{E}{X}^4=3a^2$, $\mathbb{E}{X}^{3}Y=3ac$ and $\mathbb{E}{X}^2{Y}^2=ab+2c^2$. Subtracting products of means gives, for example, $\mathrm{Var}(XY)=ab+c^2$ and $\mathrm{Cov}(X^2,Y^2)=2c^2$. In general $\mathrm{Cov}({\hat{\Sigma }}_{ij},{\hat{\Sigma }}_{kl})=({\Sigma }_{ik}{\Sigma }_{jl}+{\Sigma }_{il}{\Sigma }_{jk})/n$.

Local class field theory classifies finite abelian extensions of a local field $K$ through its multiplicative group $K^{\times }$. The Local Artin reciprocity map inducesfor each finite abelian extension $L/K$. The finite-index open subgroups of $K^{\times }$ are exactly the norm groups arising this way.

For an unramified extension of degree $n$, the norm subgroup is $⟨{\pi }^n⟩{\mathcal{O}}_K^{\times }$, where $\pi$ is a uniformizer. Under arithmetic-Frobenius normalization, the uniformizer maps to a generator of the unramified Galois group; the reciprocal normalization uses its inverse. This turns extension structure into explicit valuation and unit data.

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 $(e^T{A}^{-1}e{)}^{-1}$.

If the column player has an optimal strategy at value $v$ and a row probability vector $p$ satisfies $p^{T}A\ge v{e}^T$, then $p$ is optimal: it guarantees $v$, while the optimal column strategy prevents every row strategy from exceeding $v$.