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When dealing with infinite probability space, the probability of any particular outcome is zero. Why is this true in the uncountably infinite probability space?

An example is a Lebesgue measure on [0,1]. Why is $\mathbb{L}[a,b]=0$ for $b=a$? Is this what the author means?

Reference:
Shreve, Steven E. $\textit{Stochastic Calculus for Finance II : Continuous-Time Models}$. Springer, 2008.

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  • $\begingroup$ What do you mean by "why"? It's counterintuitive, but true. Zero doesn't mean impossible, but implausible to the highest degree (if that makes sense). $\endgroup$ – Don Thousand Sep 25 '19 at 0:25
  • $\begingroup$ The first sentence is not true, and in fact for a countably infinite probability space there always exists an outcome with positive probability. It is true that for Lebesgue measure the measure of any singleton is $0$, and that is probably what the author meant. This is a very roundabout method of proving the uncountability of the reals. $\endgroup$ – Robert Furber Sep 25 '19 at 23:33
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Not at all true. Consider the real line with Borel sigma algebra and the measure $\mu$ defined by $\mu(E)=1$ if $0 \in E$ and $0$ otherwise. Then $\mu \{0\} >0$.

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  • $\begingroup$ True, but it might be helpful to also address OP's concern about things being possible despite having $0$ possibility of occuring. $\endgroup$ – Don Thousand Sep 25 '19 at 0:25

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