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Write $Ph=\mathbb{E}h(X)$ and $P_{n}h=n^{-1}{\sum }_{i=1}^{n}h(X_i)$. A function bracket $[\ell ,u]$ contains the measurable functions $h$ satisfying $\ell (z)\le h(z)\le u(z)$ for every $z\in T$. Require its endpoints to be integrable and call $P(u-\ell )$ its $L^1(P)$ width. A sufficient condition is that, for every $\epsilon >0$, finitely many brackets of width at most $\epsilon$ cover the whole class $\mathcal{H}$. Under this condition the uniform strong law from finite L1 bracketing statesThe conclusion holds outside a common measurable null set. If the supremum is not initially known to be measurable, this formulation means pathwise convergence on a measurable probability-one event; a pointwise separable function class, including the application below, has a measurable supremum. Pointwise brackets also ensure the sample inequalities hold simultaneously over the class.

To prove the result, choose a finite $\epsilon$-cover $[{\ell }_r,u_r]$, $1\le r\le N$. If $h$ belongs to bracket $r$, monotonicity of the empirical measure and of expectation givesandConsequentlyApply the strong law of large numbers to these finitely many integrable endpoints. On a probability-one event, the maximum tends to zero. Repeat with $\epsilon =1/q$, $q\in \mathbb{N}$, and intersect the countably many probability-one events. The limiting supremum is bounded by $1/q$ for every $q$, hence is zero. This proves the uniform law of large numbers without a boundedness assumption on the class itself.

For the moment-generating function, use the empirical measure estimatorIf $t=0$, this estimator and $m$ are both identically one. Otherwise, $X\ge 0$ makes $e^{sX}$ increasing in $s$, and $e^{sX}\le e^{tX}$ provides an integrable envelope. The dominated convergence theorem shows that $m$ is continuous on $[0,t]$, hence uniformly continuous.

For any $\epsilon >0$, choose a partition $0=s_0<s_1<\cdots <s_N=t$ so that $m(s_r)-m(s_{r-1})\le \epsilon$ for every $r$. If $s\in [s_{r-1},s_r]$, then $e^{s_{r-1}z}\le e^{sz}\le e^{s_{r}z}$ for every $z\ge 0$. These endpoint functions form finitely many integrable function brackets with the required $L^1(P)$ widths. The just-proved uniform law of large numbers therefore givesBoth functions of $s$ are continuous; their supremum equals the supremum over a countable dense subset, so it is measurable. This proves uniform consistency of an empirical moment-generating function using only the observed sample.