To expand on that point—what you are refer to there as “different sizes of infinity” are different cardinalities of sets. As you note, what sorts of infinities you have to use depends on what you are trying to measure; raw cardinalities are rarely the right notion of size, here we want to think in a measure-theoretic context. But it’s worth noting that for measuring other things different systems of infinite numbers must be used; cardinalities and “infinities” should not be identified.
To expand on that point—what you are refer to there as “different sizes of infinity” are different cardinalities of sets. As you note, what sorts of infinities you have to use depends on what you are trying to measure; raw cardinalities are rarely the right notion of size, here we want to think in a measure-theoretic context. But it’s worth noting that for measuring other things different systems of infinite numbers must be used; cardinalities and “infinities” should not be identified.