I have the integral $$x=\int_R^{\infty}g(z)dF(z)$$ where $F(z)$ is the CDF of some random variable $U$. I would like to write this as an expectation with respect to $U$, i.e. $$x = \int_R^{\infty}g(z)dF(z) = \int_{-\infty}^{\infty}g(z)dF(z)1_{z\geq R}=E(g(U)1_{U\geq R})$$

The quastion what is true: $x=E(g(U)1_{U\geq R})$ or $x=E(g(U)1_{U > R})$? i.e., should I take $R$ as 1 or 0 in the indicator?

For my use to be true I see that I need to take with $>$ but I don't know how to see this directly.


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