While I am learning measure theory using ‘Measure theory and integration by de Barra, there is a definition for hereditary class of sets. There the definition is like this: “A class of sets with this property, namely that every subset of one of its members belongs to the class, is hereditary”. (Page number 94). Then every class of sets with null set as a member is hereditary. Then every sigma ring is hereditary as null set is a member. Then the smallest sigma ring which contains a given ring R and is hereditary is the sigma ring generated by the ring R. But there(in the book) it is stated that it is H(R) and it is different from the sigma ring generated by R. What is the problem or whether my understanding of the definition of hereditary class of sets is wrong? Is it, every subset of each one of its members belongs to the class? I will be extremely grateful if someone help me with examples also.
2026-04-26 06:15:18.1777184118
When we say that a class of sets is hereditary?
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When you say "Then every class of sets with null set as a member is hereditary" it seems clear you've misunderstood the definition. Not your fault, it wasn't stated very clearly. The author meant "every subset of every one of its members belongs to the class".