Definition of locally integrable function

Question

I was given two definitions:

1. Let the function $f(x)$ be defined in a interval $[a,\infty)$ we will say that $f$ is locally integrable in $[a,\infty)$ if for all $a<b$ $f$ is integrable in $[a,b]$

2. Let $f$ be defined and locally integrable in $[a,\infty)$ we will define the improper integral $\int_{a}^{\infty}f(x)dx$ to be $\lim\limits_{R \to \infty} \int_{a}^{R}f(x)dx$.
   a.if the limit exist and is finite we say that $\int_{a}^{\infty}f(x)dx$ converges and $f$ is integrable in $[a,\infty)$
   b.if the limit does not exist we say that $\int_{a}^{\infty}f(x)dx$ diverges and $f$ is not integrable in $[a,\infty)$

Is the definition 2 part b correct?

Answer 1

The second part looks right. It's actually the first one that could get some improvement. I would say:

Let the function $f(x)$ be defined on the interval $[a, +\infty)$. We say that $f$ is locally integrable if $f$ is integrable on every compact subset $[c, d] \subset [a, +\infty)$.

I think that your definition is not very clear, having multiple occurrences of $a$, but for different things.

If your confusion stems from the fact that the limit could be infinite, then know that some texts say that the integral diverges, other would write $\int_a^{+\infty} f(x)\,\mathrm dx = \infty$.