Bayesian MCMC Fitting lognorm mixture model w/constraint on population mean

I am trying to find a reference and/or guidance for fitting a mixture model to a population of data within an MCMC framwork where the global mean is constrained to a particular value, in this case 1. That is, \(E(X) = 1\) where \(f(X) = \sum \omega_i f_i(x| \theta_i)\) and the standard constraint \(\sum \omega_i = 1\) where, ideally, \(f_i\) are log normal distributions.

Any help will be appreciated.