When we perform a summation over $\mathbb{Z}$, we often encounter the convention that a sequence $u_n$ is summable over $\mathbb{Z}$ if and only if the expression $u_0 + \sum_{n=1}^{\infty} (u_n + u_{-n})$ converges. We can adopt this definition to accommodate the fact that, for example, the sequence defined by $U(k) = k$ is summable (over $\mathbb{Z}$).
Concerning integration, I worry that similar issues (multiple conventions that can cause problems with definitions) may arise:
(We consider here improper Riemann integrals)
In an exercise, I demonstrate that a specific integral $\int_{-\infty}^{\infty} f(t) dt$ is well-defined (I assume that this integral exists and is finite, and by the way, $f$ is a continuous function).
At some point, I need to justify the following claim:
$\lim_{\substack{\alpha \to \infty \ \beta \to \infty}} \int_{-\alpha}^{\beta} f(t) dt = \int_{-\infty}^{\infty} f(t) dt$, but I sense there is some justification needed for this claim.
For instance, I think that $\int_{c}^{\infty} f(t) dt$ could diverge, where $c$ is a constant (For example, $x\rightarrow x^3$).
If I consider that the integral over $\mathbb{R}$ is defined if and only if both $\int_{c}^{\infty} f(t) dt$ and $\int_{-\infty}^{c} f(t) dt$ converge, I cannot justify the above claim.
Any help to correct my confusion, or good reference that could help me, would be extremely appreciated.