What is the definition of the infimum of a family of sets? Does it mean intersection?

The RHS is a number, not a set. I assume that there is a universal set so that for all . Then if is a family of functions from to (or from to {0, 1} in this case), is probably defined pointwise: for every . So, , and it equals 1 iff for all .

It does hold under specific cases: when the sequence is mutually disjoint. The left-hand side equals 1 when belongs to any of , whereas in the right-hand side, you add 1 for each to which belongs.

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