In math, "measure" is a way of assigning a "volume" (or length, area, probability) to infinite sets. The "cardinality" of the set of numbers between 0 and 1 is the same infinity as that of the set of numbers between 0 and 2, but the standard "measure" of the latter set is twice as big. You can sum up certain types of functions defined on a continuum by "integrating" those functions values using the appropriate measure.

If there turns out to be a continuum of possible universes created by, say, a particle decay, then there's also a natural physical measure that corresponds to the probabilities we observe; the set of universes in which the particle decays before 1 half-life would be "twice as big" in some sense as the set of universes created in which the decay occurs between one half-life and two half-lifes. If someone offers to do something for you if-and-only-if a particle decays before one half life elapses, you should figure out the expected utility of a 50-50 bet, even if the reality might be that your decision is affecting two different infinities of subsequent universes.

There's a lot I'm glossing over and/or don't understand myself here (why is the probability measure the only ethical measure? lots of different-but-self-consistent measures can always be mathematically well-defined) but hopefully that at least explains the vocabulary a bit.