Hi, 


I have a question concerning integration theory I can't figure out, maybe someone can help:


Fix $\varepsilon>0$ and consider $\delta \colon [0,1] \to (0,\infty)$ measurable. Is it then true that
$$\inf\_{N\subset [0,1], \lambda(N)>\varepsilon} \int\_N \delta(t) dt > 0$$
where $\lambda(N)$ is the Lebesgue measure of $N$?


Thank you!



An infimum of integrals of a positive function.
If it wasn't the case, we would be able to find measurable sets $N\_k$ such that $\lambda(N\_k)>\varepsilon$ and the integral on $N\_k$ of $f$ is smaller than $k^{-1}$. So the sequence $\{f\chi\_{N\_k}\}$ converges in $L^1$ to $0$ (provided $f$ is integral, which can be assumed), and we extract a subsequence $\{f\chi\_{S\_k}\}$ which converges almost everywhere to $0$. So $\displaystyle\mu(\limsup\_{k\to +\infty}S\_k)=0$ and we can find an integer $j\_0$ such that $\displaystyle \mu(S\_{j\_0})\leqslant \mu\left(\bigcup\_{n\geqslant j\_0}S\_n\right)<\varepsilon/2$, a contradiction. 


