As your title suggests, the dominated convergence theorem is a powerful tool, which often we can apply to cases to where the assumption of domination is not directly verified, or nor immediate to verify .
Let's call $f_n$ the non-negative function $f_n:=e^{2mu_n}$. The first statement is a direct application of the dominated convergence theorem: since $f_n$ converges a.e. to $f:=e^{2mu}$, for every given $N$ the sequence $f_n\wedge N$ converges a.e. to $f\wedge N$, and it is dominated by the constant function $N$ (which is in $L^2(M)$, just because here $M$ has finite measure). Therefore, $f_n\to f $ in $L^2(M)$. For the other statement, note that, since, quite obviously, $f_n>N$ in the set where $f_n$ is larger than $N$,
$$\|f_n\wedge N- f_n\|_2^2=\int_{\{f_n>N\}}f_n^2dx\le \int_{\{f_n>N\}}\frac{f_n^2}{N^2} f_n^2dx\le \frac1{N^2}\|f_n\|^4_4.$$
If now the conclusion is not clear to you, write
$$\|f-f_n\|_2\le \|f-f\wedge N\|_2+\|f\wedge N-f_n\wedge N\|_2+\|f_n\wedge N-f_n\|_2\le$$$$\le \|f-f\wedge N\|_2+\|f\wedge N-f_n\wedge N\|_2+\frac1N \sup_{n\in\mathbb N}\|f_n\|^2_4$$
which is true for every $n>0$ and every $N>0$. Keep $N$ fixed and let $n\to\infty$, so by the first statement the middle term vanishes in the limit, and this yields to
$$\limsup_{n\to\infty}\|f-f_n\|_2\le \|f-f\wedge N\|_2+\frac1N \sup_{n\in\mathbb N}\|f_n\|^2_4.$$
Since this is true for all $N>0$, you can take the infimum of the RHS, which is $0$, proving $f_n\to f$ in $L^2(M)$.
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[edit] A more general statement: For a finite measure $\mu$, for $1<p\le \infty$, and for a bounded sequence $(u_n)_{n\ge0}\subset L^p(X,\mu)$ converging a.e. to $u$, one has $u_n\to u$ in $L^q$ for all $q<p$. One can prove it analogously --you can also easily reduce to the case $u_n\ge0$, $u=0$, $q=1$. (Incidentally, also $u_n\to u$ weakly* $L^p$).
Best Answer
In my understanding, these equations (like the prescribed Gauss curvature differential equation, etc) also arise from taking an infinite number of stochastic random variables, and then averaging over their behavior, which is exactly what we do when we study mean field theories.
An example of this is this paper by Michael Kiessling.