# If $P(A)=0$ then $A=\emptyset$?

Let $P$ be a probability function. It satisfied probability axioms. Can we deduce from it that if $P(A)=0$ then $A=\emptyset$ ?

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Thanks for accepting, I just updated the answer with some link. –  Ilya Jan 4 '13 at 16:18

No, e.g. if $P$ is the Lebesgue measure on $[0,1]$ then it is a probability measure, but $P(A) = 0$ for any countable $A$. One may even go further and say that $P(C) = 0$ when $C$ is a Cantor set, which is known to be uncountable. I would really advise you check out this question.
Of course, no need to use a continuous probability to find examples - you can just define a probability on $\{0,1\}$ with $P(\{0\})=P(\emptyset)=0$ and $P(\{1\})=P(\{0,1\})=1$ –  Thomas Andrews Jan 4 '13 at 16:36