I have the following theorem for integration by parts when both endpoints are finite: (Lebesgue integrals are used throughout)
Let $a\le b$ be real numbers, and $f,g$ be functions continuous on $[a,b]$ and differentiable on $(a,b)$, such that $f\cdot g'$ and $f'\cdot g$ are Lebesgue integrable on $(a,b)$. Then:
$$\int_{(a,b)}f\cdot g'=(f(b)g(b)-f(a)g(a))-\int_{(a,b)}f'\cdot g.$$
I would like to extend this theorem to the case when the interval of integration is right-unbounded. What is the most natural statement for this case? I don't think that continuous on $[a,\infty)$ and differentiable on $(a,\infty)$ is sufficient, since the direct analogue would be "continuous on $[a,\infty]$" which suggests that the behavior at $\infty$ needs to be controlled in some way – I suspect that it should be sufficient to require that the real limit of the function exist.
That said, how is something like this proven? In principle I should just be able to take a limit as $b\to\infty$ in the above theorem, but what is the appropriate theorem for taking the limit of a sequence of base sets in an integral? Something like:
If $A_n$ is a sequence of subsets of $\Bbb R$ satisfying <condition> and $f:\bigcup_n A_n\to\Bbb C$ is Lebesgue integrable on each $A_n$, and $A=\lim_n A_n$ (whatever that means), then $f$ is integrable on $A$ and $$\int_A f=\lim_n\int_{A_n}f.$$
Best Answer
If the sequence $(A_n)_n$ is increasing and $f$ is Lebesgue integrable on $E =\bigcup A_n$, then dominated convergence yields $\int_{A_n} f \to \int_E f$.
If you apply this to $A_n =(a,b_n)$ for any increasing sequence $(b_n)_n$ with $b_n \to \infty$, you get (using the fundamental theorem of calculus and the product rule)
$$ f(b_n) g(b_n) = f(a)g(a) +\int_{(a,b_n)} f'(x) g( x) + f(x) g'(x)\, dx \to f(a)g(a) +\int_{(a,\infty)} f'(x) g( x) + f(x) g'(x)\, dx. $$
Note that the limit on the right hand side is independent of the sequence $(b_n)_n$.
This implies (by rearranging)
$$ \int_{(a,\infty)} f'(x) g( x)\, dx = \lim_{b\to \infty} f(b)g(b) - f(a)g(a) - \int_{(a,\infty)} f(x) g'(x)\, dx, $$ where existence of the limit is implied by the argument above.
Observe that integrability of $f' \cdot g$ and $f \cdot g'$ was used crucially in the argument.