Suppose a continuous random variable $$L \sim F_{L}$$ where $$Pr[L = 0] = 0$$ even though $$L = 0$$ exists for some countable subset of the outcome space.
I am considering the existence of a variable
$$C \sim \text{Poisson}(L)$$
which would not be defined if $L=0$.
We could just condition that $L\neq 0$: $$C\mid L\neq0 \sim \text{Poisson}(L\mid L\neq0).$$
But in practice this doesn't make a lick of difference. Metropolis-Hastings sampling from $$\text{Poisson}(L)$$ works just fine for various choices of $F_L$ due to the almost-never occurrence of $L=0$.
Should I say that $C$ is almost-surely a random variable which follows a conditional Poisson distribution?