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Let $f(x,y)$ be a real valued function of two variables, defined for $a<y<b, c<x<d$. Assume that for each $x, f(x,.)$ is a Borel measurable function of $y$, and that there is a Borel measurable function $g: (a,b)\to$ $\Bbb R$ such that $|f(x,y)|<= g(y)$ for all $x,y $ and $\int_a^bg(y) dy <\infty.$ If $x_0\in (c,d)$ and $\lim_{x \to x_0}f(x,y)$ exist for all $y\in (a,b)$, show that $$\lim_{x \to x_0}\int_a^bf(x,y) dy = \int_a^b[\lim_{x \to x_0} f(x,y)]dy $$

(Can we use Dominated Convergence Theorem to prove this?)

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This is almost a direct application of Lebesgue's DCT as on ProofWiki. – Lord_Farin Oct 23 '12 at 12:42
up vote 0 down vote accepted

I remember a similar problem in Measure and Integral by Wheeden and Zygmund. Anyway, the problem becomes simple if you remember the following proposition:

Proposition Let $h : (c, d) \to \Bbb{C}$. Then for each $x_0 \in (c, d)$, $h(x) \to \ell$ as $x \to x_0$ if and only if for every sequences $(x_n) \subset (c, d)$ converging to $x_0$, we have $h(x_n) \to \ell$ as $n \to \infty$.

Now if you consider $h(x) = \int_{a}^{b} f(x, y) \, dy$, then the conclusion is immediate by the Lebesgue's DCT.

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@ sos440, Can I have a reference for the stated Proposition please? – tear_drops Oct 24 '12 at 11:30
@tear_drops, it is very easy to deduce, and is discussed either explicitly or tacitly in most of analysis textbooks. For example, you can refer to the p.180 of Elementary Classical Analysis by Marsden and Hoffman. – Sangchul Lee Oct 24 '12 at 13:58

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