# Integration - Measure Theory

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

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.