Suppose $(f_\alpha)$ is a net of measurable functions that converge point-wise to $f$ and that $|f_\alpha| \leq g$, for some $g \in L^1$. Does one have:
$$
\lim_{\alpha} \int f_\alpha = \int \lim_\alpha f_\alpha = \int f ?
$$
Lav Kumar Singh
Nope! This is in general not true for nets. Here's one example I can recall. Consider the directed set F of all finite subsets of [0,1] with respect to inclusion. Using the interpolation, for each E\in F, choose a continuous function f_E such that f_E(x)=1 for all x in E and \int f_E
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