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I was wondering what is the relationship between $\mathbb{E}[X^+]$ and $\mathbb{E}[X^-]$, when $\mathbb{E}$ is a sublineair expectation and $\mathbb{E}[X] = \mathbb{E}[-X] = 0, \mathbb{E}[X^2] > - \mathbb{E}[-X^2]$.

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You might want to recall (even briefly) what a sublinear expectation is, if only to avoid that people answer your question in the context of usual expectations. – Did Jan 19 '13 at 11:46

Integrating the equation $X = X^+ - X^-$ we get $E[X^+]=E[X^-]$.

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Why do you get a different answer when you use:

\begin{eqnarray*} \mathbb{E}[X^-] & = & \mathbb{E}[\max(-x,0)] \\ & = & \int^{\infty}_{0} (-x) f(x) dx \\ & = & - \mathbb{E}[X^+] \end{eqnarray*}

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First of all, you must notice that your result has to be wrong, because both $X^-$ and $X^+$ are non-negative. The mistake is that your second equality should write $\displaystyle E[\max\{-x,0\}] = \int_{-\infty}^0(-x)f(x)\,dx$ – Siméon Jan 19 '13 at 12:00

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