Let P be a probability on R. Prove that for every e > 0 there is a compact set K such that P(K) > 1 -ɛ.
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Review question 31
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- Label each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.Label each of the following statements as either true or false. Let f:AB where A and B are nonempty. Then f1(f(T))=T for every subset T of B.Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.
- 24. Let, where and are nonempty. Prove that for all subsets and of. Prove that for all subsets and of. Give an example where there are subsets and of such that. Prove that for all subsets and of. Give an example where there are subsets and of such that .Label each of the following statements as either true or false. Every least upper bound of a nonempty set S is an upper bound.Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.
- Label each of the following statements as either true or false. Let f:AB. Then f(A)=B for all nonempty sets A and B.Label each of the following statements as either true or false. If a nonempty set contains an upper bound, then a least upper bound must exist in .Let ij denote the Kronecker delta: ij=1 if i=j, and ij=0 if ij. Find the value of the following expressions. a. i=1n(j=1nij) b. i=1n(j=1n(1ij)) c. i=1n(j=1n(ij)ij) d. j=1nijjk
- Consider the probability space (Ω,A , P) where Ω = R and A is the Borel σ-algebra on R. Suppose that for any n = 1, 2, . . . , we have P((−∞, 2−n]) = 1/2 and P((−∞, −2−n]) = 1/3 − 1/(4^n) . Given the above, compute the following probability values, if possible, showing complete justifi- cation for every step. If it is impossible to compute a value precisely, provide the tightest possible bounds on it. P((−∞, 0]) P({0}) P((−∞, −1]) P((1/4, π/7]) limn→∞ P((0, n])Let M and N be nonempty bounded subsets of R and letLet A be a denumerable set. Prove that A has a denumerable subset B such that A − B is denumerable.