Arithmetic in $[0, \infty]$

1.22 Throughout integration theory, one inevitably encounters $\infty$ . One reason is that one wants to be able to integrate over sets of infinite measure; after all, the real line has infinite length. Another reason is that even if one is primarily interested in real-valued functions, the lim sup of a sequence of positive real functions or the sum of a sequence of positive real functions may well be $\infty$ at some points, and much of the elegance of theorems like 1.26 and 1.27 would be lost if one had to make some special provisions whenever this occurs.

Let us define $a + \infty = \infty + a = \infty$ if $0 \le a \le \infty$ , and

sums and products of real numbers are of course defined in the usual way.

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 cancellation laws have to be treated with some care: $a + b = a + c$ implies $b = c$ only when $a < \infty$ , and $ab = ac$ implies $b = c$ only when $0 < a < \infty$ . Observe that the following useful proposition holds:

If we combine this with Theorems 1.17 and 1.14, we see that sums and products of measurable functions into $[0, \infty]$ are measurable.