I am curious about the Dominated Convergence Theorem for a sequence of functions that converges in measure.
Theorem:
Let $(X,\mathcal{S},\mu)$ be a measure space. If $\{f_n\}, f$ are measurable, real-valued (i.e. finite) and such that $f_n \to f$ in measure and $|f_n| \leq g$ with $\int g \,\mathrm{d}\mu < \infty$, then
$$ \int f_n \,\mathrm{d}\mu \to \int f \,\mathrm{d}\mu $$
Proof:
Since $f_n \to f$ in measure, then for any subsequence of $\{f_n\}$, call it $\{f_k\}$, we have also $f_k \to f$ in measure. Now we can extract a further subsequence $\{f_{k_j}\}$ such that $f_{k_j} \to f$ almost everywhere. Applying the a.e. version of dominated convergence to this subsequence gives:
$$ \int f_{k_j} \,\mathrm{d}\mu \to \int f \,\mathrm{d}\mu $$
Now defining the sequence of real numbers $\{a_n\}$ by $a_n = \int f_n \,\mathrm{d}\mu$, we want to show that
$$a_n \to \int f \,\mathrm{d}\mu$$
But we have just shown that for any subsequence $\{a_k\}$ of $\{a_n\}$, there exists a further subsequence $\{a_{k_j}\}$ that converges to $\int f \,\mathrm{d}\mu$. Thus $a_n$ converges to $\int f \,\mathrm{d}\mu$ as well. QED.
Question:
So, nowhere in this proof did I use the fact that $\mu$ is $\sigma$-finite. However, everywhere that I look, I keep seeing this result with the condition that $\mu$ be $\sigma$-finite (e.g: Generalisation of Dominated Convergence Theorem). So I must be doing something wrong?
The only place I can think of where $\sigma$-finiteness might be required is in extracting an a.e. convergent subsequence from the “in measure” convergent sequence $\{f_k\}$. But I am pretty sure that $\sigma$-finiteness is not required to extract an almost uniformly convergent subsequence from an “in measure” convergent subsequence. And since almost uniformly convergent subsequences are also almost everywhere convergent, then I’m stumped.
Any pointers?
Some remarks:
One can prove (classical) dominated convergence theorem from Egorov’s theorem when the measure space is finite.
Then even when the measure space is not a finite union of measurable sets of finite measure, we can work on $(X_N,X_N\cap\mathcal B,\mu\mid_{X_N\cap\mathcal B})$, where $X_N=\{g>N^{-1}\}$. (we use a $2\varepsilon$-argument, choosing first $N$ such that $\int (g-g\chi_{X_N})<\varepsilon$).