Size-biased branching population measures and the multi-type $x\log x$ condition

We investigate the $x\log x$ condition for a general (Crump–Mode–Jagers) multi-type branching process with a general type space by constructing a size-biased population measure that relates to the ordinary population measure via an intrinsic martingale $W_t$. Sufficiency of the $x\log x$ condition for a non-degenerate limit of $W_t$ is proved and conditions for necessity are investigated.