The Unapologetic Mathematician

Lebesgue Measure

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

The smallest $\sigma$-ring $\mathcal{S}$ containing $\mathcal{R}$ is also the smallest one containing the collection $\mathcal{P}$ of semiclosed intervals. As it turns out, it’s also a $\sigma$-algebra. Indeed, we can write the whole real line $\mathbb{R}$ as the countable disjoint union of elements of $\mathcal{P}$.

$\displaystyle\mathbb{R}=\bigcup\limits_{i=-\infty}^\infty\left[i,i+1\right)$

and so $\mathbb{R}$ itself must be in $\mathcal{S}$. We call $\mathcal{S}$ the $\sigma$-algebra of “Borel sets” of the real line.

Our measure $\mu$ — defined on elements of $\mathcal{P}$ by $\mu(\left[b,a\right))=b-a$ — is not just $\sigma$-finite, but actually finite on $\mathcal{R}$. And thus its extension to $\mathcal{S}$ will still be $\sigma$-finite. The above decomposition of $\mathbb{R}$ into a countable collection of sets of finite $\mu$-measure shows us that the extended measure is, in fact, totally $\sigma$-finite.

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

Posted by John Armstrong | Analysis, Measure Theory | 9 Comments

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