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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 $0 \cdot \infty = \infty$?

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Have you tried checking them yourself? – Qiaochu Yuan Dec 26 '10 at 5:25
Note that the "useful proposition" that is given on the next page (namely "If $0 \leq a_1 \leq a_2 \leq \cdots$, $0 \leq b_1 \leq b_2 \leq \cdots, a_n \to a \text{ and } b_n \to b, $ then $a_nb_n \to ab$") only holds if $0 \cdot \infty$ is defined to be $0$. – Jon Dec 26 '10 at 7:06
We want the integral of the constant function 0 over a space to be 0, even if the space has infinite measure. The integral of a nonzero constant function c over a space of measure m should be cm, so Rudin's convention (0*infty = 0) makes this formula valid if c = 0 too. This is what convinced me that Rudin's convention is the right one (for integration theory). – KCd Dec 26 '10 at 10:38
@Qiaochu: I had checked and it seemed to work for $0 \cdot \infty=\infty$. Was asking to see if I was making some silly error. – Jyotirmoy Bhattacharya Dec 27 '10 at 6:59
@Jon, @KCd: Thanks. – Jyotirmoy Bhattacharya Dec 27 '10 at 6:59
up vote 7 down vote accepted

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.

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