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Let $E= \cup _{k=1}^{\infty} E_k$ where all the sets $E_k$'s are measurable and $m(E)< \infty$. Suppose $f$ is bounded and integrable one every $E_k$.

Is $f$ integrable on $E$?

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What if we consider $E_k = (\frac1{k+1},\frac1k]$ and $f = \frac1x$ over each $E_k$?

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HINT

Start with $E= (0,1]$. Note that $E$ is the union of the sets $E_n=(1/(n+1),1/n]$.

-- Now, can you find a function that is continuous and bounded on each $E_n$ but not on $E$ due to a singularity at $x=0$? How about integrability?