i couldn't arrive to the $E_x$

@IosifPinelis the proof is not detailed , he find directly $P_{x}\{y\in B]0,1]\}=\lim_{\epsilon \rightarrow 0} P_{x}\{ y\in B[\epsilon,1]\}=\lim_{\epsilon\rightarrow 0} E_xP_{B(\epsilon)}\{y\in B[0,1]\}$

@LSpice i'll do it , thank you

sorry for not being clear , i add the question

i got it , thank you so much

i add it , it existe in the reference that you shared with me too (page 12 ,a copy past from the book)

ok , i'll do that

@ThomasKojar Thank you so much

@LSpice i mean for every $x \in \mathbb{R}^2$ it give the Lebesgue measure of [a set involving $x$]

Let $B$ be a standart brownian motion , and $R$ a function defined on $\mathbb{R}^2$ and take $x$ to the Lebesgue measure of $B[0,1]\cap (x+B(t+2)-B(2)+B(1))$ \\ $Y=B(2)-B(1)$ \\ why $E(R(Y))=1/2pi\int_{\mathbb{R}^2}e^{-|x|^2}E(R(x))dx$