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

Ashot
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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.

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SBF
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    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 04 '13 at 16:36
  • @ThomasAndrews: that's true, though it seems that OP has recently started learning this topic - and I thought that the Lebesgue measure may give more "natural" and enlightening examples. – SBF Jan 04 '13 at 16:38
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    Well, if they were new to the topic of probability, then they might not even know what Lebesgue measurability means - lots of probability classes - possibly even most- do not require measure theory. – Thomas Andrews Jan 04 '13 at 16:40