Let $f \in C_0^\infty $, $g \in L^1 $ . Then $ \int_{\mathbb R^n} \int_{\mathbb R^n} f(x-y)g(y) dy dx = \int_{\mathbb R^n}\int_{\mathbb R^n} f(x-y)g(y) dx dy $holds? If so, why? ($f,g : \mathbb R^n \to \mathbb R $)
Can I change the order of the double integration?
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multivariable-calculus
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0There is something called Fubini's theorem which gives the general setting . – 2012-06-10
1 Answers
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This depends on the finiteness of the integral. Fubini's theorem can be applied:
Let $X$ and $Y$ be complete measure spaces. If $\int_{X\times Y} |f(x,y)| \ \mathrm{d}(x,y) < \infty$ then $\int_X \int_Y f(x,y) \ \mathrm{d}y \ \mathrm{d}x = \int_{X\times Y} f(x,y) \ \mathrm{d}(x,y) = \int_Y \int_X f(x,y) \ \mathrm{d}x \ \mathrm{d}y$
Since $f \in C^\infty_0$ and $g \in L^1$ I think it's safe to say that the integral in question converges.