Integral from infinity to infinity My physics professor today wrote on the blackboard:
$$ \int_{\infty}^{\infty} f(x) dx = 0 $$
for every function $f$. 
And the proof he gave was:
$$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a} f(x) dx + \int_{a}^{\infty} f(x)dx = - \int_{a}^{\infty} f(x) dx + \int_{a}^{\infty}f(x)dx = 0$$
However I'm still not convinced, for me an integral from infinity to infinity has no meaning. Therefore, what I'm asking is: does the above equations make sense? If not, are there cases where they do make sense? I'm thinking about functions that converge to 0 in $+\infty$.
EDIT: Actually, the function f considered was a density, i.e.:
$$ \int_{-\infty}^{+\infty} f(x)dx = 1 $$
and $f(x) \geq 0$ for all $x$. 

I disagree with the answers here. From Lebesgue's perspective, we can think of the integral under question as an integral over the set of all real $x$ satisfying $\infty < x < \infty$. That's an integral over the empty set, which is always 0.
