Arithmetic on $[0,\infty]$: is $0 \cdot \infty = 0$ the only reasonable choice? On page 18 of Rudin's Real and Complex analysis he defines $0 \cdot \infty = 0$ and says that "with this definition the commutative, associative, and distributive laws hold in $[0,\infty]$ without any restriction". 
What is not clear to me is whether the quoted statement is a justification of the definition or just a consequence. Wouldn't the commutative, associative, and distributive laws also hold if we define $0 \cdot \infty = \infty$? 
 A: 
It may seem strange to define $0\cdot\infty=0$.  However, one verifies without difficulty that with this definition the commutative, associative, and distributive laws hold in $[0,\infty]$ without any restriction.

The way this is worded leads naturally to your question, as though Rudin were implying that this is the main justification for defining $0\cdot\infty$ in this way.  Rather, I see this as a bonus after making the convention consistent with what happens when integrating the $0$ function or integrating over a space of measure $0$, as KCd's comment indicates.
If you ask yourself what the possibilities are, you can start by supposing that $0\cdot\infty=x$ for some $x\in[0,\infty]$, and apply the distributive law to see that $x=2x$, so that $x=0$ or $x=\infty$.  You can then verify that with either convention the commutative, associative, and distributive laws will hold, so something more is needed to motivate the choice.  Such a choice will always depend on context, and in some cases it won't be a good idea to even define $0\cdot\infty$.  However, I am not aware of a mathematical context in which the convention $0\cdot\infty=\infty$ is useful.
