# Averaging over outgoing radiation in the firewall argument

Following the AMPS paper, the Hawking radiation can be divided into an early and late part, decomposed as follows:

$$|\Psi\rangle=\sum_{i}|\psi_{i}\rangle_{E}\otimes |i\rangle_{L},$$

where $$\lbrace|i\rangle_{L}\rbrace$$ are an orthonormal basis for the late radiation. Further on, we expand $$|\psi_{i}\rangle_{E}$$ in an orthornomal basis for the early radiation, $$|\psi_{i}\rangle_{E}=\sum_{a=1}^{E}c_{ia}|a\rangle_{E}.$$ The following part is what I don't get. They say that averaging over $$|\Psi\rangle$$ with the uniform measure gives:

$$\overline{c_{ia}c^{*}_{jb}} =\frac{1}{LE}\delta_{ij}\delta_{ab}.$$

How does one get the previous relation? It makes sense to me on an intuitive level and has some sort of mathematical sense if I were to calculate it as:

$$\overline{c_{ia}c^{*}_{jb}}=\frac{\int dc_{ia}\int d_{cjb}\,c_{ia}c^{*}_{jb}}{LE},$$

and checking the parity of the integrand based on whether the indices are the same or not. However, this seems like an ad hoc solution and even if it were true, I have no way of justifying it.