There is famous statement for submodular function maximization by greedy algorithm, roughtly:
greedy submodular maximization algorithm is guaranteed to be $(1-\frac{1}{e})$-optimal approximate algorithm.
Original paper (Nemhauser 1978) and other overview article (e.g. Kraus 2014) state the above statement for submodular set function $f: 2^V \to \mathbb{R}$ where ground set $V$ is finite set.
My question is on finiteness of $V$. Reading through the proof of the above statement (e.g. page 7 of Kraus 2014), the statement can also hold when $V$ is infinite.