My book defines $$ \int_{-\infty}^{\infty}f(x) dx = \int_{-\infty}^{a}f(x)dx + \int_a^{\infty}f(x) dx$$ When:
1) $f(x)$ is integrable in $[t,t']$ for any $t, t' \in \Bbb R$ $ (t<t')$ and
2) $a \in \Bbb R$ such that both $ \int_{-\infty}^{a}f(x)dx $ and $ \int_a^{\infty}f(x) dx$ exist.
I'm wondering whether or not the first condition is necessary. If the second one holds for some $a\in \Bbb R$, then $f$ is integrable in any interval anyway, so isn't the first condition satisfied?
On
Take for example the function : $$x \mapsto x$$
Then on any compact sets $[a, b]$ we have :
$$\int_a^b x \mathrm{d}x = 0.5(\frac{b^2}{2}-\frac{a^2}{2})$$
So it’s clearly integrable on any compact sets.
Yet we have :
$$\int_{\mathbb{R_+}} x = \infty$$
So both definitions are important. So as Jose Carlos Santos said the first definition is there so that the other makes sens. Because $\int_a^{\infty} f $ is essentially defined as :
$$ \int_I f = \sup_{J \subset I} \int_I f$$
Note that we also don’t have the equivalence :
$$\lim_{x \rightarrow \infty} \int_{-x}^{x} f \text{ is integrable}\Leftrightarrow \int_{\mathbb{R}} f \text{ is integrable}$$
The first condition is there in order to assure that the second one makes sense.