The Unapologetic Mathematician

Mathematics for the interested outsider

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.

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April 19, 2010 - Posted by | Analysis, Measure Theory

9 Comments »

  1. [...] Sets and Lebesgue Measure Let’s consider some of the easy properties of the Borel sets and Lebesgue measure we introduced [...]

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  2. [...] thing that makes Lebesgue measure really special is the way that every part of the real line “looks like” every other [...]

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  3. [...] Measurable Sets The attentive reader will note that in our study of Lebesgue measure we’ve defined it on some complete -algebra . In general, there’s no reason to believe [...]

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  5. [...] For a while, we’ll mostly be interested in real-valued functions with Lebesgue measure on the real line, and ultimately in using measure to give us a new and more general version of [...]

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  6. [...] measurable functions on a measurable space would be to define a two-dimensional version of Borel sets and Lebesgue measure, and to tweak the definition of a measurable function to this space like we did before to treat [...]

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  7. [...] Upper and Lower Ordinate Sets Let be a measurable space so that itself is measurable — that is, so that is a -algebra — and let be the real line with the -algebra of Borel sets. [...]

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  8. [...] upper and lower ordinate sets and to be measurable subsets of . Now if we have a measure on and Lebesgue measure on the Borel sets, we can define the product measure on . Since we know and are both measurable, [...]

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  9. [...] Measure Algebra of the Unit Interval Let be the unit interval , let be the class of Borel sets on , and let be Lebesgue measure. If is a sequence of partitions of the maximal element of the [...]

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