= Uniform law of large numbers
{title2=ULLN}
{wiki}
Let $X_1,X_2,\ldots$ be <independent and identically distributed> with <probability law> $P$, and let $\mathcal G$ be a class of measurable $P$-integrable <functions>. Write $P_ng=n^{-1}\sum_{i=1}^ng(X_i)$ and $Pg=\mathbb E g(X_1)$. A <uniform law of large numbers> concerns
$$
\sup_{g\in\mathcal G}|P_ng-Pg|\longrightarrow0,
$$
with <measurability> of the supremum assumed. A uniform weak law uses <convergence in probability>; a uniform strong law uses <almost sure convergence>. The mode and sampling assumptions are part of the statement.
For a <finite set>[finite class], the <strong law of large numbers> for each <function> holds simultaneously and gives the uniform strong law. For infinite classes, pointwise convergence alone is insufficient. For example, let $P$ be the <uniform distribution> on $[0,1]$ and take <indicator functions>[indicators] of all <finite set>[finite subsets]. Each fixed <function> has expectation zero and <sample mean>[empirical mean] tending to zero <almost surely>. Yet for every sample the <finite set> $\{X_1,\ldots,X_n\}$ belongs to the class and its <sample mean>[empirical mean] is one. The displayed supremum is therefore one for every $n$.
Conditions such as finite <bracketing of a function class> at every positive width control this complexity. The <Glivenko-Cantelli theorem> gives the uniform strong law for <indicator functions>[indicators] of half-lines, the class determining the <empirical distribution function>.
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