I am trying to read about basic concentration of measure inequalities from Lugosi's lecture notes. ( Link to pdf ).
I cannot understand a step in the proof of Thm.7 (On Page 16 of the pdf)
The step that I cannot understand is the following. Suppose $Z= g(X_1,X_2,\ldots,X_n)$, where $(X_i)_{i=1}^n$ are independent random variables. The the author says that, it follows from the independence of $(X_i)_{i=1}^n$, that: $ \mathbb{E}[Z| (X_j)_{j=1}^{i-1}] = \mathbb{E}[\mathbb{E}[Z|X_{-i}]|(X_j)_{j=1}^i]$
Here $X_{-i} = (X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_n)$.
It seems similar to the tower property of conditional expectation, but I cannot seem to see how it is true.
The way I think of it is that $ \mathbb{E}[Z|X_{-i}]$ is a function of $X_{-i}$, and hence independent of $X_i$, and thus the conditioning it on $(X_j)_{j=1}^i$ is the same as conditioning it on $(X_j)_{j=1}^{i-1}$. But how do I show it rigorously?