Let $X$ be a compact Hausdorff space and $\mu$ be a complex Borel measure on $X$ having finite total variation. Let $B(X)$ be the set of all complex valued bounded Borel measurable functions on $X.$ Let $f_{\alpha}, f \in B(X)$ be such that $f_{\alpha} \to f$ pointwise and $\sup\limits_{\alpha} \left \{\left |f_{\alpha} (x) \right |\ |\ x \in X \right \} \lt \infty.$ Then show that $\int f_{\alpha}\ d\mu \to \int f\ d\mu.$
We have $$\left | \int f_{\alpha}\ d\mu - \int f\ d\mu \right | = \left |\int (f_{\alpha} - f)\ d\mu \right | \leq \int |f_{\alpha} - f|\ d |\mu|.$$ Now how does the given conditions conclude the result?
EDIT $:$ If we have a sequence instead of a net then we are through by the virtue of DCT. But I don't have any idea whether DCT holds for pointwise convergence of a net of measurable functions which are uniformly bounded by an integrable function.
