# Why is $M_t$ (the emphasis) helping in correcting for the state distribution in the Emphatic TD algorithm?

The book by Sutton and Barto discusses in section 11.8 that the convergence of off-policy TD function approximation can be improved by correcting for the distribution of states encountered. The section seems to be written in haste and doesn't do a good job in explaining why will $$M_t$$, the emphasis, help in getting a state distribution closer to the target policy.

My understanding of on-policy distribution is not clear at the moment. I think it is the distribution of states encountered under the target policy (the policy for which we want to state-action/state values).

The importance sampling ratio corrects for update distribution (by multiplying the correction term with the ratio), but how is $$M_t$$ helping in correcting for the state distribution?