Having a problem with this part of Measure Theory on how to set up the problem and understanding the path to the solution.
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a) Assume that $f_n \in L^1(\mathbb{R}, \mu)$ and $f_n \rightarrow 0$ almost everywhere, as $n\rightarrow \infty$ Prove,
$$ \lim_{n\rightarrow \infty} \int_{\mathbb{R}} \sin(f_n(x))e^{-x^2}d\mu(x) = 0 $$ For the above, I understand that if $g_n = \sin(f_n(x))e^{-x^2}$ then $g_n \rightarrow 0 $ but how to use this for whole space $\mathbb{R}$? The theorems that I have only works finite E.
b) Calculate
$$ \lim_{n\rightarrow \infty} \int_0^\infty \left(\frac{\sin^n(x^2)}{x^2}\right)d\mu(x) $$ if it exists. Here I don't see any convergence a sequence.
c) Consider the sequence of functions $$f_n(x) = e^{-n^4x^2}\cos^2(n\pi x)$$ and then the sequence of their sums $$ s_n(x) = \sum_{k=1}^n f_k(x) $$ Prove that $s(x) = \sum_{k=1}^\infty f_k(x)$ is integrable and that $$ \int_{\mathbb{R}} s_n(x)dx \rightarrow \int_{\mathbb{R}}s(x)dx $$ as $n\rightarrow \infty$