To keep it simple, I suggest we focus on real functions in this question.
I have heard only of 3 definitions of integrability:
- Riemann integrability
- Darboux integrability
- Lebesgue integrability
I am interested here in improper integrals i.e. integration over unbounded (infinite) intervals, so only Lebesgue is concerned. Let us assume the integration interval is $ \!R = ]-\infty,+\infty[ $.
The convention used is:
$$
\int_{-\infty}^{+\infty} f = \lim_{a\to-\infty , b\to+\infty}\int_a^b f
$$
…that is, if the limit does exist.
One may think that a function $f$ would be called "integrable" when its improper integral is finite i.e. $ |\int_{-\infty}^{+\infty} f| < +\infty $. But I believe that is an accepted definition.
However, I know that the function $f$ is called absolutely integrable or Lebesgue integrable (or $L^1$) when:
$$ \int_{-\infty}^{+\infty} |f| < +\infty $$
In signal processing, signal $s$ is said to be finite-energy when:
$$ \int_{-\infty}^{+\infty}|s|^2< +\infty $$
Seems to me its exactly the definition of a square integrable (or $L^2$) function. So far so good.
Now here is the gist. In signal processing, signal $s$ is said to be finite-power when:
$$ \lim_{T\to+\infty}\frac{1}{T}\int_{-\frac{T}{2}}^{+\frac{T}{2}}|s|^2 < +\infty $$
This is a different kind of improper integral as it is not a limit with two independent boundaries $(a,b)$ since they both depend on $T$.
So, my point is, more generally, what about functions such that:
$$ \lim_{a\to+\infty}\int_{-a}^{+a} |f| < +\infty $$
Are they classes of improper integrals in mathematics? If not, would not it have made sense to extend the concept of integrability for these functions?
EDIT1: See accepted answer about this.
EDIT2: I was actually wondering, in which cases we get $ \lim_{a\to+\infty}\int_{-a}^{+a} f $ is convergent but $ \int_{-\infty}^{+\infty} f $ is divergent. Does it only happen with odd functions?
Best Answer
The fact that the two boundaries are not independent is irrelevant, since the integrand is positive. If $f \ge 0$, then $$ \lim_{a\to-\infty , b\to+\infty}\int_a^b f = \lim_{b\to+\infty}\int_{-b}^b f $$
finite power
The main difference, though, is the factor $1/T$. In the case of "finite power", it could be that $\int_{-\infty}^{+\infty} |s|^2 = \infty$ as long as $$\int_{-\frac{T}{2}}^{+\frac{T}{2}}|s|^2$$ does not grow too fast. Growth is allowed to be at most $O(T)$ as $T \to \infty$.