Problems with Eq. (6.30)

[houp]

Describe the mistake
There is a slight formal problem in the definition of a multivariate expected value of z function g. Namely in the equation (6.30) on left hand side you write E_X[g(\bf{x})] which suggests g is a function taking an argument from \mathbb{R}^D. On the other hand, on the right hand side of the same equation you write E_{X_1}[g(x_1)] which clearly suggests g is a function taking a real argument.

Location
Please provide the

1. version Draft (2024-01-15)
2. Chapter 6, Section 6.4.1
3. Page: 187
4. Equation 6.30

Proposed solution
Clearly define what function g is and how should it be interpreted. In definition 6.3 you write g: R->R, which doesn't work in the multivariate case.