[proofplan]
We verify the hypotheses of the bracketing Donsker criterion. First, every member of $\mathcal M$ is Borel measurable and the constant function $1$ is a square-integrable envelope. The key quantitative input is the standard $L^2(P)$ bracketing entropy bound for bounded monotone functions, which gives logarithmic bracketing entropy of order $\varepsilon^{-1}$ uniformly over all probability measures $P$ on $[0,1]$. This entropy rate has finite square-root integral, so the [bracketing Donsker theorem](/theorems/9836) applies.
[/proofplan]
[step:Verify measurability and identify a square-integrable envelope]
Let $\mathcal B([0,1])$ denote the Borel $\sigma$-algebra on $[0,1]$. If $f\in\mathcal M$, then $f$ is monotone on the interval $[0,1]$, hence the set
\begin{align*}
\{x\in[0,1]:f(x)>a\}
\end{align*}
is an interval of the form $[c,1]$, $(c,1]$, $[0,1]$, or $\varnothing$ for each $a\in\mathbb R$. Therefore this set belongs to $\mathcal B([0,1])$, so $f:([0,1],\mathcal B([0,1]))\to(\mathbb R,\mathcal B(\mathbb R))$ is measurable.
Define the envelope function $F:[0,1]\to[0,\infty)$ by $F(x)=1$ for every $x\in[0,1]$.
For every $f\in\mathcal M$ and every $x\in[0,1]$,
\begin{align*}
|f(x)|\le F(x).
\end{align*}
Moreover,
\begin{align*}
\int_{[0,1]}F(x)^2\,dP(x)=\int_{[0,1]}1\,dP(x)=1<\infty,
\end{align*}
because $P$ is a probability measure. Thus $\mathcal M$ has the measurable square-integrable envelope $F$ required by the bracketing Donsker criterion.
[/step]
[step:Use the monotone bracketing entropy bound]
For $\varepsilon>0$, let $N_{[]}(\varepsilon,\mathcal M,L^2(P))$ denote the least number of $L^2(P)$-brackets of radius $\varepsilon$ needed to cover $\mathcal M$. We use the standard bracketing entropy estimate for bounded monotone functions on an interval: there is a universal constant $C>0$ such that, for every probability measure $Q$ on $([0,1],\mathcal B([0,1]))$ and every $0<\varepsilon\le 1$,
\begin{align*}
\log N_{[]}(\varepsilon,\mathcal M,L^2(Q))\le \frac{C}{\varepsilon}.
\end{align*}
This is a standard result not yet in the wiki: the $L^2(Q)$ bracketing entropy bound for uniformly bounded monotone functions.
Applying this estimate with $Q=P$, we obtain, for every $0<\varepsilon\le 1$,
\begin{align*}
\sqrt{\log N_{[]}(\varepsilon,\mathcal M,L^2(P))}
\le \sqrt{C}\,\varepsilon^{-1/2}.
\end{align*}
[guided]
The purpose of this step is to reduce the Donsker property to a concrete entropy calculation. The class $\mathcal M$ may be uncountable, so one cannot prove the result by checking finitely many functions. Instead, the bracketing entropy measures how many lower-upper function pairs are needed to trap every $f\in\mathcal M$ in $L^2(P)$.
For $0<\varepsilon\le 1$, the standard monotone-function bracketing theorem says that there is a universal constant $C>0$ such that, for every probability measure $Q$ on $([0,1],\mathcal B([0,1]))$,
\begin{align*}
\log N_{[]}(\varepsilon,\mathcal M,L^2(Q))\le \frac{C}{\varepsilon}.
\end{align*}
This estimate is uniform in $Q$, which is essential here because the theorem allows an arbitrary probability measure $P$ on $[0,1]$, including measures with atoms. We now take $Q=P$. Since the square-root function is increasing on $[0,\infty)$, the entropy estimate gives
\begin{align*}
\sqrt{\log N_{[]}(\varepsilon,\mathcal M,L^2(P))}
\le \sqrt{\frac{C}{\varepsilon}}.
\end{align*}
Equivalently,
\begin{align*}
\sqrt{\log N_{[]}(\varepsilon,\mathcal M,L^2(P))}
\le \sqrt{C}\,\varepsilon^{-1/2}.
\end{align*}
This is exactly the rate needed for the bracketing entropy integral, because $\varepsilon^{-1/2}$ is integrable at $0$.
[/guided]
[/step]
[step:Check that the bracketing entropy integral is finite]
Let $\mathcal L^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb R$.
The bracketing entropy integral associated with the envelope $F$ is
\begin{align*}
\int_0^{\|F\|_{L^2(P)}}\sqrt{\log N_{[]}(\varepsilon,\mathcal M,L^2(P))}\,d\mathcal L^1(\varepsilon).
\end{align*}
Since $\|F\|_{L^2(P)}=1$, the estimate from the previous step gives
\begin{align*}
\int_0^{\|F\|_{L^2(P)}}\sqrt{\log N_{[]}(\varepsilon,\mathcal M,L^2(P))}\,d\mathcal L^1(\varepsilon)
\le
\sqrt{C}\int_0^1 \varepsilon^{-1/2}\,d\mathcal L^1(\varepsilon).
\end{align*}
The last integral is finite and equals $2$, so
\begin{align*}
\int_0^{\|F\|_{L^2(P)}}\sqrt{\log N_{[]}(\varepsilon,\mathcal M,L^2(P))}\,d\mathcal L^1(\varepsilon)
\le 2\sqrt C<\infty.
\end{align*}
[/step]
[step:Apply the bracketing Donsker theorem]
The class $\mathcal M$ consists of measurable real-valued functions on $[0,1]$, has the measurable envelope $F$ satisfying
\begin{align*}
\int_{[0,1]}F(x)^2\,dP(x)<\infty,
\end{align*}
and has finite bracketing entropy integral in $L^2(P)$. Let $\ell^\infty(\mathcal M)$ denote the [Banach space](/page/Banach%20Space) of bounded functions $z:\mathcal M\to\mathbb R$ equipped with the supremum norm $\|z\|_\infty:=\sup_{f\in\mathcal M}|z(f)|$. Therefore the hypotheses of the Bracketing Donsker Theorem [citetheorem:9836] are satisfied. It follows that the empirical process indexed by $\mathcal M$ converges in distribution in $\ell^\infty(\mathcal M)$ to the corresponding $P$-Brownian bridge. Hence $\mathcal M$ is $P$-Donsker.
[/step]
