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Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space.

Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions:

1.$f(x,y)$ is a measurable function of $x$ and $y$ jointly, and is integrable over $y$, for almost all $x\in X$ held fixed.

2.For almost all $y\in Y, f(x,y)$ is an absolutely continuous function of $x$. (This guarantees that $\frac{\partial }{\partial x}f(x,y)$ exist almost everywhere.)

  1. $\frac{\partial }{\partial x}f(x,y)$ is locally integrable. That is, for all compact intervals $[a,b]$ contained in $X$ $$\int_{a}^{b}\int_Y|\frac{\partial}{\partial x}f(x,y)|dydx<\infty$$

Then $\int_Yf(x,y)dy$ is absolutely continuous function of $X$, and for almost every $x\in X$, its derivative exists and is given by $$\frac{d}{dx}\int_Yf(x,y)dy=\int_Y\frac{\partial}{\partial x}f(x,y)dy$$

Please help me find prove or reference.

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1 Answer 1

I suggest you have a look at this in Wikipedia it is pretty complete. And if not you have the name of what you are looking for so it should no be too difficult to find more information.

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