With and without replacement

I think that becomes key to understanding Ito.  He wants a smooth integrating variable that separates discrete events.  If we can define that, then we can chop up into small pieces and do summations. With replacement assumes empty space and Newton's grammar. Ito's tell's us what to do when we have conservation of events.

Buy, if discrete events are characterized by exchanges, locally, then there is a restriction on the events, they have a limit on mean to variance, locally.  If the restriction is met, then a contracting integration, one that converges down, will be within the same precision as the local variance.

Ito imposes us to a limited set of relationships,, we get conservation of energy, or variance. The rules are that total variance has to be accounted for, then you can use functions with the integrating variable as intermediary.