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Consider an example of Jensen's inequality in Probability. Let $X(\omega)=x\mathbf{1}_{A}(\omega)+y\mathbb{1}_{A^{c}}(\omega)$, for some $x, y\in \mathbb{R}$ such that $\mathbb{P}(A)=\lambda$. We have $$g(X(\omega))=g(x)\mathbf{1}_{A}(\omega)+g(y)\mathbb{1}_{A^{c}}(\omega)$$

Why above equality is right? I cannot understand that it because it is the same with $g(\alpha X+\beta Y) =g(\alpha)X+g(\beta)Y$. How to prove it?

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For any $\omega$ either $ \omega \in A$ or $ \omega \in A^{c}$. In the first case both sides of the equation are $g(x)$ and in the second case both sides are $g(y)$ so the equation is true.

$g(\alpha X+\beta Y)=g(\alpha)X+g(\beta)Y$ is not true in general.

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