dealing with truncation
I have been given the info that a sequence of functions fn satisfies all of the criteria needed for the monotone convergence theorem.
Now I take (g)k to be the truncation of a function g at height k. So the truncation of fn would be
(fn)k(x) = min(k,fn(x))
also akn is the integral of (fn)k(x) on (k,-k)
I am trying to show that akn converges to the integral of fn(x) as k tends to infinity and akn converges to the integral of (f)k(x) as n tends to infinity.
But I am really struggling to break this down?
Now I take (g)k to be the truncation of a function g at height k. So the truncation of fn would be
(fn)k(x) = min(k,fn(x))
also akn is the integral of (fn)k(x) on (k,-k)
I am trying to show that akn converges to the integral of fn(x) as k tends to infinity and akn converges to the integral of (f)k(x) as n tends to infinity.
But I am really struggling to break this down?
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