Negligible Set in Uncountably Infinite Space

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.

• 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). – Don Thousand Sep 25 '19 at 0:25
• 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. – Robert Furber Sep 25 '19 at 23:33

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