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Say f is a non-negative, integrable function over a measurable set E. Suppose

[tex]

\int_{E_k} f\; dm \leq \epsilon

[/tex]

for each positive integer [itex]k[/itex], where

[tex]

E_k = E \cap [-k,k]

[/tex]

Then, why is it true that

[tex]

\int_E f\; dm \leq \epsilon \quad ?

[/tex]

I know that

[tex]

\bigcup_k E_k = E

[/tex]

and intuitively it seems reasonable, but I don't know how to prove it.

[tex]

\int_{E_k} f\; dm \leq \epsilon

[/tex]

for each positive integer [itex]k[/itex], where

[tex]

E_k = E \cap [-k,k]

[/tex]

Then, why is it true that

[tex]

\int_E f\; dm \leq \epsilon \quad ?

[/tex]

I know that

[tex]

\bigcup_k E_k = E

[/tex]

and intuitively it seems reasonable, but I don't know how to prove it.

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