Stochastic(Markov) matrix

Question

We say that a matrix $P $ = $(P_{ij} : i, j \in I)$ where $I$ is a countable set, is stochastic if every row $(P_{ij} : j \in I)$ is a distribution.

What do we mean by $distribution$ here?

Answer 1

All "distribution" means here is that each $P_{ij}$ is non-negative (that is, $P_{ij} \geq 0$) and $$ \sum_{j \in I} P_{ij} = 1 $$ In general, a (discrete) distribution is any countable set of non-negative values that sum to $1$.