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Answer :
Final answer:
The lim sup of an indicator function corresponds to the indicator function of the lim sup of the sets, where the correct answer is B) Supremum. Lim sup represents the least upper bound of subsequential limits, and in the context of indicator functions, it describes the set of elements that appear infinitely often.
Explanation:
The question deals with the concept of limit superior (lim sup) in the context of indicator functions. Lim sup is a term used in mathematics to describe the supremum (least upper bound) of all subsequential limits of a sequence. The lim sup of an indicator function indicating a set is indeed the indicator function of the lim sup of the sets. To answer the multiple-choice question: Lim sup of indicator function is the indicator of lim sup, the correct option is B) Supremum.
For example, consider a sequence of sets (An) where each set presents a condition defining membership. The indicator function of each set An (denoted IAn) is a function that takes a value of 1 for elements inside the set and 0 for elements outside the set. The lim sup of the sequence of sets (An) is the set of elements that belong to infinitely many An. The indicator function of this lim sup set (Ilim sup An) will take a value of 1 for elements that are in the lim sup set and 0 otherwise. Thus, it can be seen that the lim sup of the sequence of indicator functions matches the indicator function of the lim sup of the sets.
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