## Integrable Functions

Now we can define what it means for a general real-valued function (not just a simple function) to be integrable: 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 limit

But how do we know that this doesn’t depend on the sequence ?

We recall that we defined

which must be measurable for any measurable function . This is the only part of the space that matters when it comes to integrating ; clearly we can see that

since is zero everywhere outside .

Now, if both and converge in measure to , then we can define to be the (countable) union of all the and . Just as clearly, we can see that

where is the indefinite integral of , and is the indefinite integral of . Then if we use to define the integral of we get

while if we use we get

But we know that since and both converge in measure to the same function, the limiting set functions and coincide, and thus . The value of the integral, then, *doesn’t* depend on the sequence of integrable simple functions!

[...] « Previous | [...]

Pingback by Properties of Integrable Functions « The Unapologetic Mathematician | June 3, 2010 |

[...] Convergence Properties of Integrals Okay, we’ve got our general definition of integrable functions, and we’ve reestablished a bunch of our basic properties in this setting. Let’s [...]

Pingback by Mean Convergence Properties of Integrals « The Unapologetic Mathematician | June 4, 2010 |

[...] is an Integral Zero? We’ve got some interesting results about when integrals come out to be [...]

Pingback by When is an Integral Zero? « The Unapologetic Mathematician | June 7, 2010 |

[...] we use the definition of integrability, and we take the supremum over finite, pairwise disjoint collections . But [...]

Pingback by An Alternate Approach to Integration « The Unapologetic Mathematician | June 18, 2010 |

[...] Indefinite Integral So, after all our setup it shouldn’t be surprising that we take an integrable function and define its indefinite [...]

Pingback by The Jordan Decomposition of an Indefinite Integral « The Unapologetic Mathematician | June 29, 2010 |