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For sigma-finite measures $\nu$ and $\mu$, there is a unique decomposition $\nu ={\nu }_{\mathrm{ac}}+{\nu }_{\mathrm{s}}$ in which ${\nu }_{\mathrm{ac}}\ll \mu$ and ${\nu }_{\mathrm{s}}\perp \mu$.

For sigma-finite measures, $\nu \ll \mu$ implies $\nu (A)={\int }_A(d\nu /d\mu )d\mu$ for a nonnegative measurable density unique almost everywhere.

Two measures are mutually absolutely continuous when they have exactly the same null sets.

If $\mu (\Omega )<\infty$, then $\mu \ll \nu$ exactly when for every $\epsilon >0$ there is $\delta >0$ such that $\nu (A)<\delta$ implies $\mu (A)<\epsilon$. For necessity, a contrary sequence with $\nu (A_n)<2^{-n}$ and $\mu (A_n)\ge \epsilon$ has tail unions decreasing to a $\nu$-null limsup; continuity from above for finite $\mu$ contradicts absolute continuity.

Let $h:\mathbb{R}\to \mathbb{R}$ be continuous, strictly increasing, and satisfy Lusin condition N. Thendefines a measure on the Lebesgue measurable sets, and $\nu \ll \lambda$. Moreover,for $a\le b$.

A function $f$ has Lusin's condition $(N)$ when it maps every Lebesgue-null set to a Lebesgue-null set.

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar{z}$, the vortex equations are

A finite abelian group combines a finite set of reversible operations with commutativity. All finite cyclic groups qualify, but direct products of groups supply examples that need more than one generator of a group. The Fundamental theorem of finitely generated abelian groups says that every finite abelian group is a direct product of groups of finite cyclic groups; the finite case has no infinite cyclic factors.^[ref]

These factors distinguish different ways to have the same total number of states. A single four-position dial gives $C_4$, while two independent binary switches give $C_2\times C_2$. Both have four states and commutative composition, yet only the dial permits a single operation whose repetition visits all four states. The following Cayley tables make that distinction visible without relying on the labels alone.

An element $a$ of an abelian group is infinitely divisible when, for every positive integer $n$, there is an element $b$ with $nb=a$. Compactness can create a nonzero infinitely divisible element in a nonstandard model even when the original group has none.

A complex in an abelian category is a sequence $\cdots \to A_{n+1}\stackrel{d_{n+1}}{\to }{A}_n\stackrel{d_n}{\to }{A}_{n-1}\to \cdots$ with $d_n{d}_{n+1}=0$.

For the two-stage implicit Runge-Kutta method withthe stability function isFor $z=x+iy$,The denominator has no zero in the closed left half-plane, so the method is A-stable for every real $a$.

A square game with antisymmetric payoff matrix has value zero.

An optimal mixed strategy guarantees the value of the game against every opposing pure strategy.

For every real $m\times n$ payoff matrix $A$ of a two-player zero-sum game, the minimax theorem stateswhere ${\Delta }_m,{\Delta }_n$ are the players' probability simplices. Both outer optima are attained. The row player's security level payoff is the largest payoff it can guarantee against all opposing choices; the column player can hold that payoff to the same number $v$, the value of a zero-sum game. For optimal $x^{\ast },y^{\ast }$,These are exactly the Nash equilibrium inequalities under the row-maximizes, column-minimizes convention. Conversely, a Nash equilibrium of this two-player zero-sum game supplies both guarantees. This equality explains why independent worst-case optimization solves the opposing incentives in this special class.

Allowing mixed strategies is essential. For matching pennies, $A=\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$. Restricting both players to pure strategies gives ${\max }_i{\min }_j{A}_{ij}=-1$ and ${\min }_j{\max }_i{A}_{ij}=1$. Allowing mixed strategies closes the gap: each fair mixture guarantees zero, so $v=0$. These guarantees hold even if the opponent knows the mixture's probabilities, provided the opponent does not observe the sampled choice before choosing.

John von Neumann proved the two-player zero-sum game result in Zur Theorie der Gesellschaftsspiele, Mathematische Annalen 100, 295–320 (1928); the original article's footnote 7 explicitly discusses the equality of max–min and min–max. Nash's ^[ref] states that existence of a Nash equilibrium is equivalent to this result in the two-player zero-sum game case, and extends existence to all finite games.

A matrix game is a finite two-player zero-sum game whose entry $A_{ij}$ is the row player's payoff when the players choose row $i$ and column $j$.

The union of the three coordinate axes in ${\mathbb{A}}^3$ and a union of three distinct lines through the origin in ${\mathbb{A}}^2$ are not isomorphic. Their origins are intrinsically the unique points common to all three irreducible components, but their Zariski tangent spaces there have dimensions three and two respectively.

Forin characteristic other than two,The variety has dimension two. Its tangent dimension is two when both coordinate pairs are nonzero, three when exactly one pair vanishes, and four at the origin.

Since $T_{(x,y)}(X\times Y)=T_{x}X\oplus T_{y}Y$, if $Y$ is smooth thenIn particular, dimensions add.

The smooth locus consists of points $P$ for which $\dim T_{X,P}=\dim X$. For an irreducible variety over a perfect field it is a nonempty Zariski-open subset and therefore dense.

For an affine variety, its dimension is the minimum of $\dim T_{X,P}$ over its points. A point is smooth when its tangent dimension attains this minimum and singular otherwise.