The motivation is to see when I can use the Dominated Convergence Theorem or the Monotone Convergence Theorem when all I know is the convergence of the function pointwise in a dense subset.
Let $(X,\mathcal{B}(X))$ be a measurable space with the Borel sigma algebra. Let $D\subset X$ be a dense subset of X. Let $f_n$ and $f$ be measurable functions such that, for every $d\in D$, $f_n(d)\to f(d)$. Let $\mu$ be any arbitrary measure supported by the space. I´m interested in conditions such that
$$\lim_{n\to\infty}\int_Xf_nd\mu=\int_Xfd\mu$$
One I have in mind is when $f$ and $f_n$ are continuous, so as the set $D$ is dense, I can always construct a sequence within $D$ tending to any point $x\in X\setminus D$ and pass through the limit to x. I'm tempted to say the argument follows when the set of discontinuities of $f_n$ and $f$ is of measure zero too.
Are any other conditions someone recall to extend the pointwise convergence from the dense subset to the entire set, or to make work the DCT or MTC when we only know the pointwise convergence in the dense subset?
Thanks!
Let $g_n:=f_n-f$. For the DCT, it suffices to assume that $|f_n|\le h$ for all $n\ge 1$ with $h\in L_1$ and the sequence of functions $\{g_n\}$ is pointwise equicontinouos $\mu$-a.e. Then for $x\in A$ with $\mu(A^c)=0$ and any $\epsilon>0$, there exists $x'\in D$ s.t. $$ \limsup_{n\to\infty}|g_n(x)|\le \limsup_{n\to\infty}|g_n(x)-g_n(x')|+\lim_{n\to\infty}|g_n(x')|< \epsilon, $$ which implies that $g_n\to 0$ a.e.
The above result holds if one replaces the pointwise equicontinuity with a slightly weaker condition: for almost all $x\in X$ and every $\epsilon>0$, there exists $\delta_x>0$ s.t. $$ \limsup_{n\to\infty}|g_n(x)-g_n(y)|<\epsilon $$ whenever $|x-y|<\delta_x$.