Let $(S,\mathcal S)$ be a measurable space, let $P$ be a probability measure on $(S,\mathcal S)$, and let $\mathcal F$ be a class of [measurable functions](/page/Measurable%20Functions) $f:S\to\mathbb R$. Suppose that $\mathcal F$ has a measurable envelope $F:S\to[0,\infty]$ satisfying $|f(x)|\le F(x)$ for $P$-a.e. $x\in S$ and every $f\in\mathcal F$, and suppose \begin{align*} \int_S F(x)^2\,dP(x)<\infty. \end{align*} For $\varepsilon>0$, let $N_{[]} (\varepsilon,\mathcal F,L^2(P))$ be the least cardinality of a finite collection of brackets $[l,u]$ covering $\mathcal F$, where $l,u:S\to\mathbb R$ are finite-valued measurable functions in $L^2(P)$, where $l\le u$ $P$-a.e., and where \begin{align*} \left(\int_S (u(x)-l(x))^2\,dP(x)\right)^{1/2}<\varepsilon. \end{align*} Set \begin{align*} H_{[]} (\varepsilon,\mathcal F,L^2(P)):=\log N_{[]} (\varepsilon,\mathcal F,L^2(P)). \end{align*} Assume \begin{align*} \int_{(0,\infty)}\sqrt{H_{[]} (\varepsilon,\mathcal F,L^2(P))}\,d\mathcal L^1(\varepsilon)<\infty. \end{align*} Let $(\Omega,\mathcal A,\mathbb P)$ carry i.i.d. $S$-valued random variables $X_1,X_2,\dots$ with common distribution $P$. For every measurable $h:S\to\mathbb R$ with $h\in L^2(P)$, define \begin{align*} Ph:=\int_S h(x)\,dP(x) \end{align*} and \begin{align*} G_n(h):=n^{-1/2}\sum_{i=1}^{n}\{h(X_i)-Ph\}. \end{align*} Then $\mathcal F$ is $P$-Donsker: the maps $G_n|_{\mathcal F}$ converge in outer distribution in $\ell^\infty(\mathcal F)$ to the tight $P$-Brownian bridge $G_P$, the centred Gaussian process indexed by $\mathcal F$ with covariance \begin{align*} \operatorname{Cov}(G_P(f),G_P(g))=P(fg)-Pf\,Pg \end{align*} for all $f,g\in\mathcal F$. All suprema and weak-convergence assertions are interpreted with outer probability and outer expectation when measurability is not known.