Behaviour of Probability in Limits

Question

Suppose we have a probability distribution over $\mathbb{R}$. Let $a, \epsilon \in \mathbb{R}$ be arbitrary. Suppose we define $$b = \text{inf}\{ b' : b'<a, \mathbb{P}\big[x \in [b', a]\big] < \epsilon\}.$$ Can we say that $\mathbb{P} \big[x \in (b, a]\big] < \epsilon $

Answer 1

Not with the strict inequality. For instance, let $X\sim U([0,1])$ be the uniform random variable on $[0,1]$. Let $a,\epsilon=1$. Then clearly $b=0$, but $\mathbb{P}[X\in (0,1]]=1$.

It is true with the weaker conclusion that $\mathbb{P}[X\in (b,a]]\leq \epsilon$. This is immediate from continuity of measure.