A question concerning Ulam's Theorem from Oxtoby's "Measure and Category"

Question

I am reading the following theorem from Oxtoby's Measure and Category

Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if it is equal to zero for every one-element subset.

My question is: what does it mean for a set $A$, to vanish identically? I mean, identically to what?

Thank you! Shir

Answer 1

"What does it mean for a set $A$, to vanish identically?" --- It doesn't mean anything, but nothing in the material you quoted mentions a set vanishing identically. The closest it comes is in the statement of the theorem, where it refers to a measure $\mu$ vanishing identically. That means that $\mu(S)=0$ for all $S$. (Quite generally, a function vanishes identically if its value is $0$ for all inputs in its domain.)