I tried to think about the question by this approach: what if there is a sequence of measures $\mu_n$ which uniformly converge to $\mu$ and we had all the hypothesis of Lebesgue's Dominated Convergence Theorem like:

Suppose that ${f_n}$ is a sequence of measurable functions where each $f_n$ is measurable regarding to $\mu_n$, that $f_n \to f$ point-wise almost everywhere as $n\to \infty$, and that $|f_n|\to g$ for all n, where $g$ is integrable.

then do we have the result of LCT here? i mean, is $f$ integrable and do we have the equation below? $$\int f d\mu=\lim_{n\to \infty} \int f_nd\mu_n$$

  • $\begingroup$ What do you mean by uniform convergence of measures? By the way, what you write is not exactly the assumption of the classical dominated convergence theorem: we should have $\sup_n |f_n|$ integrable. $\endgroup$ – Davide Giraudo Nov 14 '16 at 13:47
  • $\begingroup$ @DavideGiraudo convergence of the measure is the exact vague part for me too. i thought maybe it can be think of like $\lim \mu_n E $ when n goes to infinity is exactly $\mu E$. and also since g is integrable the assumption $sup_n|fn|$ be integrable is satisfied. $\endgroup$ – Parisa Mahmoudi Nov 15 '16 at 6:38

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