What techniques are there to evaluate the Lebesgue integral of an integrable function $f$ on the whole $\mathbb{R}$?
I am following a class on Measure Theory and these questions make my head go blank, whenever there is not a obvious sequence of simple functions which converges to $f$ I am clueless. The convergence theorems are known, but is this the only possible route of attack?
For example: $\int_{\mathbb{R}}\left|2ye^{-y^2}\right|dy$
2 Answers
The fundamental theorem of calculus still holds for the Lebesgue integral. In particular, if $F$ is absolutely continuous on $[a,b]$, then
$$ F(b) - F(a) = \int_a^b F'(t)\,dt. $$
So, you can integrate the usual way, by finding an antiderivative, and then examining the limit as $a \to \infty$, $b \to -\infty$ using your convergence theorems (i.e. multiply by $\chi_{[-N,N]}$).
A bounded function on an interval is Riemann integrable iff the set of discontinuities has Lebesgue measure zero.
Using Riemann sums and this fact shows that a bounded Riemann integrable function on an interval is Lebesgue integrable and the integrals are the same. Hence if you know how to Riemann integrate a function, you know the Lebesgue integral.
