Monotone convergence - help required

[woundedtiger4]

Hi all,

http://www.scribd.com/doc/100079521/Document-1

Actually, I am trying to learn monotone convergence theorem, and I am stuck at one specific point, on the first page it says that ∫-∞→∞ f_n(x)dx = 1 for every n but the almost everywhere limit function is identically zero, what does it mean? how come the first is equal to 1 and the other is equal to zero?

Thanks in advance.

 

[tiny-tim]

hi woundedtiger4!

… on the first page it says that ∫-∞→∞ f_n(x)dx = 1 for every n but the almost everywhere limit function is identically zero, what does it mean? how come the first is equal to 1 and the other is equal to zero?

i don't understand why you're asking

the limit (as n -> ∞) is obviously 0 (everywhere except x = 0)

(and the integral happens to be 1, for all n, though that's less easy to prove)

 

[alan2]

Draw yourself a picture. It might help you see what's going on. You have a sequence of normal densities with decreasing variance. For each n, the density must integrate to 1 because it's a normal density. But as n increases, the density gets narrower (by virtue of decreasing variance) and taller. So the sequence of densities converges to 0 everywhere except at x=0 where it's getting taller with increasing n. This should all be clear algebraically but sometimes a picture helps to clarify the concept.

 

[woundedtiger4]

Draw yourself a picture. It might help you see what's going on. You have a sequence of normal densities with decreasing variance. For each n, the density must integrate to 1 because it's a normal density. But as n increases, the density gets narrower (by virtue of decreasing variance) and taller. So the sequence of densities converges to 0 everywhere except at x=0 where it's getting taller with increasing n. This should all be clear algebraically but sometimes a picture helps to clarify the concept.

thanks a tonne

 

[blue_raver22]

Hello,

Thank you for reaching out for help with the monotone convergence theorem. I understand that you are confused about the statement on the first page of the document you provided, where it says that the integral of the function f_n(x) is equal to 1 for every n, but the almost everywhere limit function is identically zero.

First, let me explain the monotone convergence theorem. This theorem states that if you have a sequence of non-negative, measurable functions f_n(x) that are increasing (or decreasing) pointwise, then the limit of the sequence will also be a measurable function and the integral of the limit function will be equal to the limit of the integrals of the individual functions. In other words, if we let F(x) be the limit function, then ∫ f_n(x) dx → ∫ F(x) dx as n → ∞.

Now, going back to the statement in the document, it is saying that for every n, the integral of f_n(x) is equal to 1. This means that each individual function has an integral of 1. However, the almost everywhere limit function, which is the limit of the sequence, is identically zero. This means that for almost every point x, the value of F(x) is equal to zero. This may seem contradictory, but it is possible for a sequence of functions to have integrals that converge to a non-zero value, while the limit function itself is equal to zero at almost every point.

I hope this helps clarify your confusion. If you have any further questions, please do not hesitate to ask. Keep up the good work in learning the monotone convergence theorem!

Best,