## Sequences of Measurable Functions

We let be a sequence of extended real-valued measurable functions on a measurable space , and ask what we can say about limits of this sequence.

First of all, the function is measurable. The preimage is the union of the countable collection , while the preimage is the intersection of the countable collection . And so both of these sets are measurable, and we can restrict to the case of finite-valued functions.

So now let’s use our convenient condition. Given a real number we know that if and only if for some . That is, we can write

Each term on the right is measurable since each is a measurable function, and so the set on the left is measurable. Thus we conclude that is measurable as well.

Similarly, we find that the function is measurable.

Now the functions

are also measurable. Indeed, in proving that is measurable we can use the exact same technique as above to prove that the inner supremum is measurable; it doesn’t really depend on the supremum starting at or higher. And then the outer infimum is exactly as before. Proving is measurable is similar.

Now we can talk about pointwise convergence of a sequence of measurable functions. That is, for a fixed point we have the sequence which has some limit superior and some limit inferior . If these two coincide, then the sequence has a proper limit . But one of our lemmas tells us that the set of points where any two measurable functions coincide has a nice property: has a measurable intersection with every measurable set. And thus if we define the function on this subspace of for which the limit exists, the resulting function is measurable.

[...] functions! That is, given any measurable function we can find a sequence of simple functions converging pointwise to [...]

Pingback by Simple and Elementary Functions « The Unapologetic Mathematician | May 11, 2010 |

[...] Convergence Almost Everywhere Okay, so let’s take our idea of almost everywhere and apply it to convergence of sequences of measurable functions. [...]

Pingback by Convergence Almost Everywhere « The Unapologetic Mathematician | May 14, 2010 |

[...] sequence of measurable functions and concludes that is measurable (along with the inequality), but we already know that the limit inferior of a sequence of measurable functions is measurable, and so the integrable [...]

Pingback by Fatou’s Lemma « The Unapologetic Mathematician | June 16, 2010 |

I am reading simple measurable functions, but I couldn’t locate the convergence of simple measureable functions precisely , can anyone help me in that area?

Comment by manju | September 23, 2012 |