Fubini's Theorem

By David Hodge (P1262) on Tuesday, November 2, 1999 - 12:17 pm :

What is Fubini's Theorem and how is it related to being able to interchange the order of evaluating Summations and integrals in an equation?

By Dave Sheridan (Dms22) on Tuesday, November 2, 1999 - 03:21 pm :

This is a theorem in measure theory, which justifies swapping two integrals. I'll assume that you know measure theory, and explain from there. If you don't, please let me know and I'll give you a brief overview (but it's rather complex...)

Let P and Q be finite measures (so we'll assume they're probability measures). If f is a measurable function then

P(Q(f))=Q(P(f))

whenever either is finite. If you rewrite this as an integral, you get

òòf dQ dP = òòf dP dQ

whenever either is finite.

Now a summation is merely an integral with respect to an atomic measure, so this covers the case of swapping the sum and integral signs.

If you need any help on specific problems, please let me know; if this has been too technical for you then I can simplify it instead.

-Dave

By David Hodge (P1262) on Wednesday, November 3, 1999 - 11:27 am :

I haven't really done any measure theory, this question came up because I was evaluating a big sum of factorials by evaluating them to Beta functions i.e. a sum of Beta functions (which themselves are integrals) I have been told that Fubini's Theorem justifies doing the summation first then integrating the expression obtained which comes out as just B(2n+1,2)+B(2,2n+1).
When can Fubini's Theorem be invoked?

By Dave Sheridan (Dms22) on Thursday, November 4, 1999 - 06:42 pm :

Basically, if you know that the integral of the sum is finite, you can exchange the two. Similarly, if you know the sum of the integral is finite, you can exchange the two. So if you swap the two and end up with something finite, this justifies it. But you must prove that one or the other is finite.

-Dave

By David Hodge (P1262) on Friday, November 5, 1999 - 10:22 am :

Thanks.