The interpretation of $0 \cdot \infty$ When I read books about measure theory (in particular Folland's Real analysis), authors note that, sometimes, they assume $0\cdot\infty = 0$. I think, it is obvious because if you sum infinitely many zero, you will obviously get zero. Then, why this is called an interpretation? Conversely, when I think about zero times $\infty$, the result should be again zero. I do not understand the interpretability of this statement. Thanks.
 A: As remarked in the comments, $0\cdot \infty$ makes no sense as an ordinary product. There are situations where it is useful to agree that $0 \cdot \infty =0$: the best known situation is abstract measure theory.
But this is a convention that cannot preserve the continuity of the product, as the well-known examples of indeterminate limits show.
A: This is one of several situations where a common definition leads to a situation that behaves badly with respect to limits.  Here the ancient notion of product, namely that $ab$ is the area of a rectangle whose sides have lengths $a$ and $b$, leads to $0\cdot\infty=0$, and this works fine when dealing with lengths and areas, as in measure theory.  For another example, the simplest definition of exponentiation of natural numbers, namely that $a^b$ is the number of functions from a $b$-element set to an $a$-element set, leads to $0^0=1$, and this works fine when we deal with combinatorics of finite sets.  
Once limits enter the picture, these definitions require some caution.  When calculating limits, one frequently uses "facts" like, if $\lim_{x\to q}f(x)=a$ and $\lim_{x\to q}g(x)=b$, then $\lim_{x\to q}(f(x)g(x))=ab$ and $\lim_{x\to q}f(x)^{g(x)}=a^b$, and these become false with the definitions in the preceding paragraph.  My own inclination is to say "OK, multiplication and exponentiation aren't continuous at these points, so you'd better apply these "facts" only at places where continuity holds."  (In other words, don't use continuity at points of discontinuity.)  Many people (including especially calculus students) get in the habit of just using these "facts" while ignoring the need to check any hypotheses.  To prevent these people from getting wrong answers, other people (including especially the authors of calculus texts) have adopted the policy of calling things like $0\cdot\infty$ and $0^0$ indeterminate forms.  This is a way of warning people loudly to avoid using continuity in these cases.  That is, instead of saying $0^0=1$ but watch out when using this in limits, one says $0^0$ is undefined, so you can't evaluate it in limits.  
Since people usually learn calculus early in their mathematical careers, they get indoctrinated with the "knowledge" that $0\cdot\infty$ and $0^0$ (and other such things) are undefined.  As a result, when a measure theorist wants to use the natural definition, which makes $0\cdot\infty=0$, or when a combinatorialist wants to use the natural definition that makes $0^0=1$, they must point this out explicitly.  The bold ones will say "the correct definition gives this"; the more timid ones, who want to remain at peace with the calculus books, will say "we adopt this convention."
