If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is $f \in L^p(0,T)$?

Question

If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is it necessarily true that $f \in L^p(0,T)$?

I don't see why not since the only point we have a problem may be at 0, but that is a null set. By the way doesn't this mean also that $L^p_{loc}(0,T) = L^p(0,T)$?

Answer 1

This is not true. Take for example $p=1$ and the function $f_\epsilon(x)=\frac{1}{x}$ for $x>\epsilon$ and 0 else. Then $f_\epsilon$ is in $L^1(\epsilon,T)$ with norm equal to $ln(\epsilon)-ln(T)$. Bug the norm diverges as $\epsilon$ goes to 0. Hence $\frac{1}{x}$ is not in $L^1(0,T)$.