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Unfortunately I've forgotten some notation I came across recently. Can someone clarify what is the meaning of what is shown in the red box?enter image description here

The book is Probability Essentials by Jacod-Protter.

ervx
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Cure
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  • @kimchilover Then is it the same as $f(X_n)-f(Y_n)=1$ if $||X_n-Y_n||>\epsilon$ and 0 $f(X_n)-f(Y_n)=0$ if $||X_n-Y_n||\leq\epsilon$ ? – Cure Oct 19 '17 at 21:42

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$\|X_{n}-Y_{n}\|>\varepsilon$ is notation for the set $A:=\{x\in\Omega:\|X_{n}(x)-Y_{n}(x)\|>\varepsilon\}$. Thus, $1_{\{\|X_{n}-Y_{n}\|>\varepsilon\}}$ is the indicator function of the set $A$. I.e., $$ \begin{aligned} 1_{\{\|X_{n}-Y_{n}\|>\varepsilon\}}(x)&= \begin{cases} 1 &\text{ if }x\in A\\ 0 &\text{ if }x\notin A \end{cases} \\ &= \begin{cases} 1 &\text{ if }\|X_{n}(x)-Y_{n}(x)\|>\varepsilon\\ 0 &\text{ if }\|X_{n}(x)-Y_{n}(x)\|\leq\varepsilon \end{cases} \end{aligned} $$

Thus, $E\{|f(X_{n})-f(Y_{n})|1_{\{\|X_{n}-\|Y_{n}\|>\varepsilon\}}\}$ is the expected value of the function $g(x)$, where

$$ \begin{aligned} g(x)= \begin{cases} |f(X_{n}(x))-f(Y_{n}(x))| &\text{ if }\|X_{n}(x)-Y_{n}(x)\|>\varepsilon\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 &\text{ if }\|X_{n}(x)-Y_{n}(x)\|\leq\varepsilon \end{cases} \end{aligned} $$

ervx
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  • So it is a product? Like $E{|f(X_n)-f(Y_n)|}1_{||X_n-Y_n||>\epsilon}=0$ if $||X_n-Y_n||\leq \epsilon$ and $E{|f(X_n)-f(Y_n)|}1_{||X_n-Y_n||>\epsilon}=E{|f(X_n)-f(Y_n)|}$ if $||X_n-Y_n||> \epsilon$. – Cure Oct 19 '17 at 23:11
  • I have updated my answer. I think it is, in effect, what your are saying. – ervx Oct 20 '17 at 00:31