Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
Thanks for accepting, I just updated the answer with some link. – Ilya Jan 4 '13 at 16:18
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.