recently I started studying the domain that is analysis 2. More specifically the last few days I have been busy with the topic of Lebesgue integrals. Whilst making an exercise I got stuck with trying to approximate some sequence of functions to make use of the dominated divergence theorem so that I could interchange the limit and integral sign. The problem is the following: Let $t>0$, now I want to interchange the limit and integral of the following expression
$$\lim_{t \to \infty}\int_{0}^{\infty} \frac{1-(\cos(t) +y\sin(t))e^{-ty}}{1+y^2} dy$$
According to the dominated convergence theorem, in order to bring this limit inside of the integral I need to find a positive integrable function $g: [0;\infty[ \rightarrow [0;\infty]$ for which it holds that
$$|f_t(x)| \leq g(x)$$
where $f_t(x)$ denotes the function inside the above integral for some t. This needs to hold for every $y \in [0;\infty[$ and every $t>0$. I found the following function for which this inequality holds, namely $g(y) = \frac{2+y}{1+y^2}$, however this function is not integrable on the interval $[0;\infty[$. Does anyone have a tip for a better function g which is indeed integrable? Any help would be greatly appreciated :))
Using dominated convergence theorem to switch limit and integral
lebesgue-integralreal-analysis
Best Answer
Since $t\to\infty$, it is enough to show $|f_t(x)|\leq g(x)$ for every $0\leq x<\infty$ and every $t>1$. This gives$$|1-(\cos{t}+x\sin{t})e^{-tx}|\leq2+xe^{-x}\leq2+M$$ since $xe^{-x}$ is bounded on $[0,\infty)$ as mentioned by geetha290krm