Q:

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow 0}\dfrac{r}{x}=+\infty$, for $r\in (0,+\infty)$. And, $0\times \infty=0$ (in which $0$ means absolute zer0).
In measure theory, we extend non-negative real axis to $[0,+\infty]=[0,+\infty)\cup{\{r_0=+\infty}\}$ (I used of a definition came in Terence Tao's book in here, page $xi$ - "... with an additional element
adjoined to it, which we label $+∞$"). So what is the value of $0\times +\infty$ (in which $+\infty = r_0$ and $0$ is the absolute value)?   
Detailed answers with supportive proofs would be really appreciated.  

A:

Michael Hardy has already explained why one wants to say $0\cdot(+\infty)$ is undefined (or indeterminate) in some contexts and 0 in at least one other context, namely integration.  I think the value 0 is appropriate in geometric contexts too.  Imagine a "rectangle" of 0 height but infinite width; it's essentially just a line, and its area, which ought to be length times width, is 0.  The value 0 also makes sense in some combinatorial contexts.  Think of $a\cdot b$ as the number of ordered pairs you can form with the first component taken from a given set of size $a$ and the second component taken from a given set of size $b$. If the first set is empty ($a=0$), then there are no such ordered pairs, even if the second set is infinite.  (Admittedly, in such a situation, I'd rather use an infinite cardinal for the size of the second set, rather than just $+\infty$.)
I'll go out on a limb and claim that 0 is always the right answer except when you're dealing with a limiting process, with a factor approaching 0 and another factor approaching $+\infty$, as in the first part of Michael Hardy's answer.  In other words, the formula $0\cdot(+\infty)=0$ works fine as long as you don't get the idea that the multiplication operation is continuous at $(0,+\infty)$.

