## 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.

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

Pingback by Borel Sets and Lebesgue Measure « The Unapologetic Mathematician | April 20, 2010 |

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

Pingback by Lebesgue Measure and Affine Transformations « The Unapologetic Mathematician | April 22, 2010 |

[...] 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 [...]

Pingback by Lebesgue Measurable Sets « The Unapologetic Mathematician | April 23, 2010 |

[...] is a Borel set contained in . The difference set contains no point of , since if this happened we’d have [...]

Pingback by Non-Lebesgue Measurable Sets « The Unapologetic Mathematician | April 24, 2010 |

[...] 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 [...]

Pingback by Measurable (Extended) Real-Valued Functions « The Unapologetic Mathematician | April 30, 2010 |

[...] 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 [...]

Pingback by Adding and Multiplying Measurable Real-Valued Functions « The Unapologetic Mathematician | May 7, 2010 |

[...] 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. [...]

Pingback by Upper and Lower Ordinate Sets « The Unapologetic Mathematician | July 20, 2010 |

[...] 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, [...]

Pingback by The Measures of Ordinate Sets « The Unapologetic Mathematician | July 26, 2010 |

[...] 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 [...]

Pingback by The Measure Algebra of the Unit Interval « The Unapologetic Mathematician | August 25, 2010 |