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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 $ ?

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Thanks for accepting, I just updated the answer with some link. –  Ilya Jan 4 '13 at 16:18
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up vote 14 down vote accepted

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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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
    
@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. –  Ilya Jan 4 '13 at 16:38
    
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 4 '13 at 16:40
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