Equivalence between $\sigma$-additivity and a certain condition, on a certain set of equivalence classes I'm facing a certain part of a problem, and I don't know how to solve it. The background is the following:
In a probability space $(\Omega,\mathcal{B},\mathbb{P})$, two sets $A,B$ are equivalent if $\mathbb{P}(A\bigtriangleup B) = 0$. This can define equivalence classes as follows:
$$A^\# = \{ B \in \mathcal{B} : \mathbb{P}(A\bigtriangleup B) = 0 \},$$
and we can define a probability in the set of equivalence classes as $\mathbb{P}^\#(A^\#) = \mathbb{P}(A)$. I know that this is a metric space with metric
$$d(A^\#,B^\#) = \mathbb{P}(A\bigtriangleup B).$$
Furthermore, I already proved that $\mathbb{P}^\#$ is uniformly continuous on this set. I need to prove the following:
$$ \mathbb{P} \text{ is } \sigma\text{-additive} \iff [\mathcal{B} \ni B_n \downarrow \emptyset \implies d(B_n^\#,\emptyset^\#) \to 0 ].$$
I think the $\implies$ part is easy: if I take sets $B_n \downarrow\emptyset$, then
$$d(B_n^\#,\emptyset^\#) = \mathbb{P}(B_n\bigtriangleup \emptyset) = \mathbb{P}(A_n) \to \mathbb{P}(\emptyset) = 0,$$
where the previous convergence is true due to $\sigma$-additivity.
Sadly, I have no clue to prove the $\impliedby$ part. Any help will be appreciated. Thank you very much in advance.

Let $A_k$ be a sequence of pairwise disjoint sets.  Set $B_n = \bigcup_{k=n}^\infty A_k$.  Then $B_n \downarrow \emptyset$ so by assumption $d(B_n^\#, 0^\#) \to 0$ which is to say $\mathbb{P}(B_n) \to 0$.  Now by finite additivity, for any $n$ you have
$$\mathbb{P}\left(\bigcup_{k=1}^\infty A_k\right) = \sum_{k=1}^{n-1} \mathbb{P}(A_k) + \mathbb{P}(B_n)$$
and now pass to the limit.
