Let $Y(1)$ and $Y(0)$ be the potential outcomes under the treatment and the control. $T$ represents the treatment status. We let
$ATT=E(Y(1)-Y(0)|T=1)$
Next, we let $e(x)$ denote the propensity score. One type of propensity score weighting scheme is: if treated, they get a weight of 1; if not treated, they get a weight of $\frac{e(x)}{1-e(x)}$. We then let
$\Delta=\frac{TY}{1}-\frac{(1-T)Y}{1/\frac{e(x)}{1-e(x)}}$
Now how can we we show that
$E(\Delta)=E(\frac{TY}{1}-\frac{(1-T)Y}{1/\frac{e(x)}{1-e(x)}})=E(Y(1)-Y(0)|T=1)$. So that we know $E(\Delta)$ is unbiased for ATT and weights $(1,\frac{e(x)}{1-e(x)})$ is for ATT.?
I can show that
\begin{align} E(I(T=1)Y) &=E(I(T=1)Y(1)) \, \hspace{1cm} Y(1)=Y \, \text{because of consistency}\\ \end{align}
Now I have a hard time to understand how can we get \begin{align} E(I(T=1)Y(1))=E(Y(1)|T=1) \end{align}
If so, it follows that \begin{align} E(I(T=1)Y(0))=E(Y(0)|T=1) \end{align}
Any suggestion will be appreciated.