Consider the usual measurable space of real number $( \mathbb{R}, \mathcal{B}(\mathbb{R}))$. My question is:

Is there an application $\mu$ on $\mathcal{B}( \mathbb{R}) \rightarrow [0,+\infty]$ such that :

i) $\mu$ is finitely additive

ii) $\mu( \mathbb{R}) < \infty$

iii) $\exists (A_n) \subset \mathcal{B}( \mathbb{R}) $ that:

a) $ A_n \subset A_m \forall n<m $

b) $ \lim_{n\rightarrow \infty} \mu(A_n) < \mu( \cup_{n} A_n)$