Let $\xi$ be a real value random variable and $x\in \mathbb{R}$, we define the function $$\mathbb{1}_{[x,x+h]}(y):=\left\{\begin{array}{ll}1 & \mbox{if }y\in[x,x+h]\\0 & \mbox{if }y\notin[x,x+h] \end{array}\right. .$$ In this context, the following question have come to my mind:
Suppose that $\xi$ is a continuous variable with distribution $\mathbb{P}$ (which is usually unknown).
Is there any probability measure $\mathbb{Q}$ such that $$\lim_{h\rightarrow 0}\mathbb{E}_{\mathbb{Q}}\left(\mathbb{1}_{[x,x+h]}(\xi)\right)\neq 0 \:\:?$$
Remark: Note that $\mathbb{Q}$ can not be $\mathbb{P}$ because $\xi$ is a continuous random variable.