# How to determine point of zero measure?

Today in our physics lecture, our Prof told us during some calculation that for $x\rightarrow0$

$f(x)\rightarrow\frac{1}{x^2}$

which was easily understandable from the context and our previous calculations. However he then continued with the integral

$\int_0^\infty 4\pi x^2 f(x)\, dx$

and told us that it had had measure $0$ for $x\rightarrow0$. From there one we gave the $x=0$ case a special treatment, since it had physical relevance. My question is now, what does it mean to have a measure of $0$ at some point? Would the intgral

$\int_0^\infty x\, dx$

have measure 0 at $x=0$? What measure would the integral

$\int_0^\infty \frac{1}{x}\, dx$

have at $x=0$ and even more, what measure would the integral

$\int_{y(T)}^\infty \frac{1}{x}\, dx$

have if $y(T)$ is a function approaching zero for $T\rightarrow0$. How do I calculate measure? Although it probably sounds like a homework question, these questions arose during discussions with other students.