- #1
tjkubo
- 42
- 0
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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