  We study a standard two-source model for common randomness (CR) generation in
which Alice and Bob generate a common random variable with high probability of
agreement by observing independent and identically distributed (i.i.d.) samples
of correlated sources on countably infinite alphabets. The two parties are
additionally allowed to communicate as little as possible over a noisy
memoryless channel. In our work, we give a single-letter formula for the CR
capacity for the proposed model and provide a rigorous proof of it. This is a
challenging scenario because some of the finite alphabet properties, namely of
the entropy can not be extended to the countably infinite case. Notably, it is
known that the Shannon entropy is in fact discontinuous at all probability
distributions with countably infinite support.
