Show, that if f n → f and f n → g is μ-convergent, then...

Mylee Underwood

Mylee Underwood

Answered

2022-07-05

Show, that if f n f and f n g is μ-convergent, then f = g almost everywhere on X
Hint
Use the fact, that:
{ x X : f ( x ) g ( x ) } = m = 1 { x X : | f ( x ) g ( x ) | 1 m }
So, I don't know how to use that hint. μ convergent means (correct me if I'm wrong), that
f n f  is  μ  convergent μ ( { x X : ε lim n | f n ( x ) f ( x ) | > ε } ) = 0
So I don't see it how the hint should be used.
It is not written what kind of measure our μ is though, usually when there's nothing written we assume it's a Lebesgue measure, but I don't know if that has to be the case here

Answer & Explanation

billyfcash5n

billyfcash5n

Expert

2022-07-06Added 17 answers

I would guess μ-convergent means the sequences converge in measure, which means for every ε > 0, μ ( x X : | f n ( x ) f ( x ) | > ε ) 0 as n . One way to proceed is note that
| f g | | f f n | + | f n g | ,
and use this to show that for every ε > 0,
μ ( | f g | > ε ) μ ( | f f n | > ε 2 ) + μ ( | f n g | > ε 2 ) 0  as  n .
Another approach is to use the fact that convergence in measure implies a subsequence converges almost everywhere. You can use this to get a subsequence f n k f a.e., and then since f n k g in measure, there is a subsequence f n k j g a.e. Therefore f = g a.e, both being a.e. limits of f n k j .

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