The following question should be part of the questions I recently asked here Prove or disprove a claim related to $L^p$ space

If $g \in L^p(\Omega, \lambda)$ where $\Omega$ is a bounded subset of $\mathbb{R^n}$, $p>1$ and $\lambda$ is the Lebesgue measure. By Holder's inequality, we know that for any measurable set $E \subset \Omega$, $$\int_E |g| d \lambda \le ||g||_p \lambda (E)^{\frac{p-1}{p}}$$.

Now the question is, if there exits a constant C, such that for any measurable set $E \subset \Omega$, $$\int_E |g| d \lambda \le C \lambda (E)^{\frac{p-1}{p}}$$ Does this imply $g \in L^p(\Omega, \lambda)$?

Here is my partial work. I tried to use the the duality of $L^q$ space by contradiction or the fact that simple functions are dense to prove this characterization, but I cannot control the constant. I think I need a powerful elementary inequality. Or maybe this characterization is not true.

Any comments would be appreciated. Thanks!