In the 2003 survey paper on MCMC methods by Andrieu et al, there is a section on importance sampling. More specifically, in the section included above it is claimed that
$\sum_{i=1}^N f(x^{(i)})w(x^{(i)}) $ is unbiased.
However, according to my own calculation
$E[\sum_{i=1}^N f(X^{(i)})w(X^{(i)})]= \sum_{i=1}^N \int f(x)w(x)q(x)dx =N \int f(x)p(x)dx=NI(f) $.
What went wrong? Both q(x) and p(x) are distributions so there is no funny business to do with normalizing constants.
it does look like he has forgotten a $1/N$. Just a simple typo.