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