Definir:
$$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid FL(x)\geq \alpha\rbrace}=VaR\alpha(L)$$
Quiero probar que:
$$ES\alpha = \frac{1}{1-\alpha}\mathbb{E}[\mathbb{1}{\lbrace{ L\geq q\alpha(L)\rbrace}}\cdot L] \overset{!!!}{=}\mathbb{E}[L\mid L\geq q\alpha(L)] $$
Me quedo atascado como:
$$\mathbb{E}[\mathbb{1}{\lbrace{ L\geq q\alpha(L)\rbrace}}\cdot L]= \mathbb{E}[\mathbb{E}[\mathbb{1}{\lbrace{ L\geq q\alpha(L)\rbrace}}\cdot L\mid L\geq q\alpha(L)]] = \mathbb{E}[\mathbb{1}{\lbrace{ L\geq q\alpha(L)\rbrace}}\cdot\mathbb{E}[L\mid L\geq q\alpha(L)]\ ]$$
Ahora me gustaría usar ese %-%-%, pero no sé cómo proceder.