Proposition: If $f$ is Lebesgue integrable function on measurable set $E$, and $\int_E |f|dx = C_1 < \infty$, then $f$ is essentially bounded, i.e. there exists a constant $C_2$ such that, $$ |f(x)| \le C_2, \quad \text{a.e. in } E. $$
Does the above proposition make scence. If it does, how to prove it?
If it is not, why? Moreover, Lebesgue integrable function should be finite almost everywhere. What extra condition should be added to make sure that it is bounded almost everywhere?
No. Let $E=(0,1)$ and $f(x)= \frac{1}{\sqrt{x}}.$