Suppose . Show that the integral

converges for a.e. .

**Solution.** Let

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.

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