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Both a sigma algebra and the powerset are sets of subsets. However:
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The powerset \(\; 2^S \;\) of a set \(\; S \;\) is the set of all subsets. There is only one powerset.
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A sigma algebra \(\; \Sigma \;\) is just a set of subsets that fulfills certain closure conditions. There are many possible sigma algebras that can be derived from a single set \(\; S \;\).
NOTE Since the powerset is the set of all subsets, and a sigma algebra is a set of subsets, it follows that \(\; \Sigma \subseteq 2^S \;\). Note the possible equality: in fact, the powerset is a sigma algebra! This is because it contains all subsets, so it contains both the full set and the empty set, as well as the unions of all subsets, thus fulfilling all the conditions of a sigma algebra.