I am preparing for an exam in Measure and Integration Theory (Lebesgue Integration). As far as I know my professor prefers to ask students solving explicit integrals which can be solved using the main convergence theorems (Monotone convergence, Fatou, differentiation Therorems etc.) and the theory about product measures.
What I mean by explicit integrals, are the following exercises,
Compute, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ or show that $F:(0,\infty)\rightarrow \mathbb R$ given by, $$ F(t)=\int_0^{\infty}\frac{\sin(x)}{x}e^{-tx}dx$$ is well defined, and compute if possible $F'$.
Thus I wanted to ask if someone knows a source where I can get such exercises from. Or maybe you know "nice" integrals which can be solved using Lebesgue Integration Theorie.
In the most books I checked the exercises where rather focussed on abstract theorie with proves.
Thanks in advance!
In Folland's Real Analysis: Modern Techniques and Their Applications, he has a few. Problems $27 - 31$ (some of which have multiple parts) of chapter $2$ seem to fit your description. If you know about $n$-dimensional Lebesgue measure, problems $55 - 61$ of the same chapter may also be of interest.