Use the monotone convergence theorem to calculate $\int_{-\infty}^{+\infty}e^{-|x|}d\bar{\lambda} $

Question

For a particular Lebesgue exercise, I am required to work out the integral $\int_{-\infty}^{+\infty}e^{-|x|}d\bar{\lambda} $ using the monotone convergence theorem. I am finding it difficult to start, mainly as the monotone convergence theorem generally involves a sequence of functions that converge to a particular function, but in this case, we are given a singular function.

Answer 1

I think what you are asked to do is write the integral as the limit of $\int I_{(-n,n)} e^{-|x|}dx$ (which is permissible by Monotone Convergence Theorem) so we get the answer as $\lim 2\int_0^{n} e^{-x}dx =\lim 2(1-e^{-n})=2$.