Radon-Nikodym: Integrability?

Question

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$.

Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall E\in\Sigma)\implies\kappa(E)=\int_Ehd\lambda \quad(\forall E\in\Sigma)$$ for some representative $0\leq h\leq L$.

Now, what about integrability: $$f\in L(\kappa)\iff f\in L(\lambda)$$

So far I got that: $$\int|f|d\kappa=\int|f|\cdot|h|d\lambda\leq L\int|f|d\lambda$$ which proves one inclusion...

Answer 1

This is not true: $$f\in L^1(\kappa)\iff f\in L^1(\lambda).$$

However, from $\kappa(E)\le c\lambda(E)$, you can deduce that

$$ f\in L^1(\lambda)\quad\Longrightarrow\quad f\in L^1(\kappa). $$