What is the meaning of "$<\infty$"?

Question

Does the following notation just mean that the integral isn't positively infinite? $$\int _a^b f(x)\,\mathrm{d}x < \infty$$

Does it not also mean that the integral converges to some finite value?

Does it not also mean that the integral doesn't diverge to (minus) infinity? Or, do I, for the last purpose, have to emphasize by writing

$$\left|\int _a^b f(x)\,\mathrm{d}x\right| < \infty$$

Answer 1

I view this as what one might call "physicist's notation", which does not have a precise definition attached but whose meaning can be inferred easily. It can be understood to mean

It is not the case that $\int_a^b f(x)\,\mathrm dx=\infty$,

or perhaps

$\int_a^b f(x)\,\mathrm dx$ has a value and that value is not $\infty$,

(because one would not say that $\int_0^\infty\cos x\,\mathrm dx<\infty$.) I have always seen it used in association with an integral that is nonnegative, so there is no need to emphasize that $\int_a^b f(x)\,\mathrm dx\ne-\infty$.

I would say it is rare to encounter an integral where you really need to convince the reader that it does not go to infinity in either direction. But if you do, it would be clearest to just state in words that $\int_a^b f(x)\,\mathrm dx$ is finite.