I am given the definition of the Lebesgue Measure:
Lebesgue Measure $\mathcal P$ on the Borel $\sigma$-algebra $\mathcal B[0,1]$: $\mathcal P[[a,b]] = b - a$ for all $0 \leq a \leq b \leq 1$
The definition states the following: $\mathcal P[[a,a]] = a - a = 0$. Therefore $\mathcal P[(a,b)] = \mathcal P[[a,b]] = b - a$
Why can the last statement be postulated? The Lebesgue Measure of the open interval should be infinitesimal smaller than the one of the closed interval. Where do I make a mistake in this thinking?