MTH411
Question: Let G be a non-empty open set. Then the sum of the length of all its component intervals, /( m(G)=∑_k▒〖m(I_k ),where I_k 〗 /) are the component intervals on G, is called
Answer: a measure
Question: A family of subsets of an arbitrary set X that contains and is closed under complementation, under the formation of countable unions and under the formation of countable intersections is called
Answer: a /( σ -/) algebra
Question: Let F be a bounded closed set. Which of the following is true?:
Answer: The measure of F is the least upper bound of the measures of the closed sets contained in F
Question: Let F be a bounded closed set contained in the open interval I. one of the following is true:
Answer: /( m(F) = m(I)-m(I/F) /)
Question: Let the open interval J be the union of finite or denumerable family of open sets, that is /( J= ⋃_k▒G_k ,/) , then
Answer: ) /( m(J) ≤∑_k▒(G_k ) /)
Question: The greatest lower bound of the measures of all bounded open sets containing a bounded set E is called
Answer: outer measure of E.
Question: Assume that /( G_1,G_2 /) are open sets such that /(G_1 ⊆ G_2, /) , then
Answer: /(m(G_1 ) ≤ m(G_2 ) /)
Question: Let X be any nonempty set and let M be a /( σ -/) algebra. Then the pair (X,M) is called
Answer: a measurable space
Question: If F is a bounded closed set, then F is
Answer: non-negative
Question: The /( σ -/) -algebra on /( R^n /) generated by the collection of open subsets of /( R^n /)
Answer: a Borel /( σ -/) algebra
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