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tripes3h

Answered question

2022-07-12

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Suppose

E

is a measurable set and

m (E) < + \infty

{f_{k}}

is a sequence of non-negative measurable functions. If

lim_{k \to + \infty} \int_{E} \frac{f_{k} (x)}{1 + f_{k} (x)} d x = 0,

prove that

f_{k}

converge to 0 in measure.

If we can prove

\frac{f_{k} (x)}{1 + f_{k} (x)} \to 0

a.e.

x \in E

, then the problem is easy to solve. Is it correct? If so, how to prove that?
Appreciate any hint or help!

Nathen Austin

Beginner2022-07-13Added 14 answers

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document

\int_{E} \frac{f_{k} (x)}{1 + f_{k} (x)} d x \geq \int_{E (ϵ)} \frac{f_{k} (x)}{1 + f_{k} (x)} d x

where

E (ϵ) = {x \in E : f_{k} (x) > ϵ}

. Hence

\int_{E} \frac{f_{k} (x)}{1 + f_{k} (x)} d x \geq \frac{ϵ}{1 + ϵ} m (E (ϵ))

. This shows that

m (E (ϵ)) \to 0

k \to \infty

for any

ϵ > 0

which means

f_{k} \to 0

in measure on

E

.
I have used the fact that

\frac{y}{1 + y}

is an increasing function on

(0, \infty)

y > ϵ

implies

\frac{y}{1 + y} > \frac{ϵ}{1 + ϵ}

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