On computing a conditional expectation for countable-co-countable sigma-algebras

Question

Let $S=[0,1]$ equipped with its Borel $\sigma$-algebra ${\cal B}. $ Assign on it a purely atomic probability measure, $P $. Let ${\cal A} $ be the sub-$\sigma$-algebra generated by the countable and co-countable sets. Now, take $B=[0, 1/2]. $ What is $E_{P}[I_B \mid {\cal A}] $ ? Since there is a countable set $A =\{x\in S: P({x}) > 0\} $ and ${\cal A} $ contains all singletons, I am tempted to say that $E_{P}(I_B \mid {\cal A})(x) =I_B(x) $ for all $x \in A. $ However, I do not seem to be able to make this argument rigorous. Any help would be appreciated.