Let $\mu: \Sigma \to \overline{\mathbb{R}}$ a signed Radon measure and $\lambda : \Sigma \to [0,+\infty]$ a positive Radon measure, where $\Sigma$ is the Borel $\sigma$-algebra generated by the topology of the real line, denoted by $\tau$. Write $\mu = \mu^+ - \mu^-$ and suppose that
$$ |\mu(A)| \le \lambda (A), \quad \forall A \in \tau $$
Using this, is it possible to prove that $|\mu(A)| \le \lambda(A)$ for every measurable set $A \in \Sigma$?
Here in this link, there is a proof for the case where $\mu$ is positive and there is equality rather than inequality.