Suppose . Show that the integral
converges for a.e. .
Let be the ball centered at the origin of radius
We can show that
(The constant can be computed – it’s actually – but the exact value is immaterial.) Hence,
Since this is finite, we must have almost everywhere on Since we can repeat this for any (choose a countable sequence of such tending to ) we can say that is finite almost everywhere on Hence the integral that defines converges absolutely for almost every Note the importance here of having the function be locally integrable – in fact, uniformly locally integrable.