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I was in a class and the professor said he knew of no applications (in a proof or elsewhere) of these two generalizations of Fatou's lemma and Lebesgue's dominated convergence theorem that I write below.

I want to look for such an application, but it's very difficult to look up anything about them when they aren't "named" theorems.

These theorems come from Royden's Real Analysis, Chapter 11, Section 4.

Proposition 1: Let $(X, \mathscr{B})$ be a measurable space, $<\mu_n>$ a sequence of measures that converge setwise to a measure $\mu$, and $<f_n>$ a sequence of nonnegative measurable functions that converge pointwise to the function $f$. Then

$$\displaystyle\int f d{\mu} \leq \liminf \displaystyle\int f_n d{\mu_n}.$$

Proposition 2: Let $(X, \mathscr{B})$ be a measurable space and $<\mu_n>$ a sequence of measures on $\mathscr{B}$ that converge setwise to a measure $\mu$. Let $<f_n>$ and $<g_n>$ be two sequences of measurable functions that converge pointwise to $f$ and $g$. Suppose that $|f_n| \leq g_n$, and that

$$\lim \displaystyle\int g_n d{\mu_n} = \displaystyle\int g d{\mu} < \infty.$$


$$\lim \displaystyle\int f_n d{\mu_n} = \displaystyle\int f d{\mu}.$$

I am aware that Vitali's convergence theorem is a generalization of Lebesgue's dominated convergence theorem, but it does not seem to be the same generalization.

Does this generalization have a name? If not, do you know any applications of it?

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I'm not sure if you're interested in this, but anyway it contains some applications, and the results are simply called "generalized Fatou" and "generalized Lebesgue", there. There are some pointers to the literature in there. – t.b. Dec 8 '11 at 18:08

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