Let $(\Omega, \mathcal{M}, \mu)$ be a measure space and let $f\ge 0$ be a measurable function on $\Omega$. Suppose that $f$ satisfies the following properties:
- For all $\varepsilon > 0$ there exists a $\delta > 0$ such that for any measurable subset $A\in \mathcal{M}$ such that $\mu(A)<\delta$, we have $\int_A f(x)\, \mu(dx) < \varepsilon\;$
- For all $\varepsilon > 0$ there exists a measurable subset $B\in \mathcal{M}$ such that $\mu(B)<\infty$ and $\int_{\Omega \setminus B} f(x)\, \mu(dx) < \varepsilon.$
Question Does it follow that $f\in L^1(\Omega)$?
Clearly it is sufficient to consider only the special case in which $\Omega$ is a probability space. Then I am able to answer affirmatively if $\Omega$ is non-atomic by appealing to the decomposition result explained in this post by Byron Schmuland. That is, since $\Omega$ supports a random variable with uniform $(0, 1)$-distribution, for a fixed value of $\varepsilon$ (say, $\varepsilon = 1$) we can decompose $\Omega$ into a finite disjoint union of sets of measure smaller then $\delta$ and then conclude by means of assumption 1.
But this seems too complicated for a result which I feel should be trivial. What am I overlooking?
Thank you.