Lebesgue Measure

So we’ve identified a measure on the ring of finite disjoint unions of semiclosed intervals. Now we want to apply our extension and completion theorems.

The smallest -ring containing is also the smallest one containing the collection of semiclosed intervals. As it turns out, it’s also a -algebra. Indeed, we can write the whole real line as the countable disjoint union of elements of .

and so itself must be in . We call the -algebra of “Borel sets” of the real line.

Our measure — defined on elements of by — is not just -finite, but actually finite on . And thus its extension to will still be -finite. The above decomposition of into a countable collection of sets of finite -measure shows us that the extended measure is, in fact, totally -finite.

But our measure might not be complete. As the smallest -algebra containing , might not contain all subsets of sets of -measure zero. And thus we form the completions of our -algebra and of our measure. We call the -algebra of “Lebesgue measurable sets”, and is “Lebesgue measure” (remember, it’s pronounced “luh-BAYG”). In fact, the incomplete measure on Borel sets is also often called Lebesgue measure.