## Convergence in Measure I

Suppose that and all the (for positive integers ) are real-valued measurable functions on a set of finite measure. For every we define

That is, is the set where the value of is at least away from the value of . I say that converges a.e. to if and only if

for every .

Given a point , the sequence fails to converge to if and only if there is some positive number so that for infinitely many values of . That is, if is the set of points where doesn’t converge to , then

Of course, if is to converge to a.e., we need . A necessary and sufficient condition is that for all . Then we can calculate

Our necessary and sufficient condition is thus equivalent to the one we stated at the outset.

We’ve shown that over a set of finite measure, a.e. convergence is equivalent to this other condition. Extracting it a bit, we get a new notion of convergence which will (as we just showed) be equivalent to a.e. convergence over sets of finite measure, but may not be in general. We say that a sequence of a.e. finite-valued measurable functions “converges in measure” to a measurable function if for every we have

Now, it turns out that there is no metric which gives this sense of convergence, but we still refer to a sequence as being “Cauchy in measure” if for every we have

[…] proposition we started with yesterday shows us that on a set of finite measure, a.e. convergence is equivalent to convergence in […]

Pingback by Convergence in Measure II « The Unapologetic Mathematician | May 21, 2010 |

[…] and Algebra Unlike our other methods of convergence, it’s not necessarily apparent that convergence in measure plays nicely with algebraic operations on the algebra of measurable functions. All our other forms […]

Pingback by Convergence in Measure and Algebra « The Unapologetic Mathematician | May 21, 2010 |

[…] some things about this notion of convergence. For one, a sequence that is Cauchy in the mean is Cauchy in measure as well. Indeed, for any we can define the […]

Pingback by The L¹ Norm « The Unapologetic Mathematician | May 28, 2010 |

[…] and are mean Cauchy sequences of integrable simple functions, then they’re both also Cauchy in measure, which implies that they each converge in measure to some function. If they converge to the same […]

Pingback by Indefinite Integrals and Convergence II « The Unapologetic Mathematician | June 1, 2010 |

[…] a function is integrable if there is a mean Cauchy sequence of integrable simple functions which converges in measure to . We then define the integral of to be the […]

Pingback by Integrable Functions « The Unapologetic Mathematician | June 2, 2010 |

[…] of characteristic functions must converge in mean to some function . But mean convergence implies convergence in measure, which is equivalent to a.e. convergence on sets of finite measure, which is what we’re […]

Pingback by Completeness of the Metric Space of a Measure Space « The Unapologetic Mathematician | August 9, 2010 |

among all the modes of convergence, which one is the strongest and which one is the weakest?

Comment by cyprian | February 25, 2011 |

In what context? Not all modes of convergence are defined at the same time.

Comment by John Armstrong | February 25, 2011 |