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1. Let u be the counting measure on P(N). Find a decreasing sequence of sets Ak...
6. In this problem, u is Lebesgue measure on R, while v is counting measure on...
6. In this problem, u is Lebesgue measure on R, while v is counting measure on N. Let =u X v be the product measure and let f: R *N → R be given by f(x, n) = 1 + (21 x)? Compute S f d). Remember that S itu du = arctan u + C.
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak...
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
(Exercise 4.13, reordered) Given a series ΣΧί ak, let 8,-Ση-i ak. Σχί ak is Cesaro summable if S1 + 82 +... +Sn lim n-+...
(Exercise 4.13, reordered) Given a series ΣΧί ak, let 8,-Ση-i ak. Σχί ak is Cesaro summable if S1 + 82 +... +Sn lim n-+o converges. (a) Give an example of a series Σ00i ak that is Cesaro sum mable but not convergent (b) Prove that if 1 ak converges, then it is Cèsaro summable. Hint: Say the sequence of partial sums sn → L. Try to prove that =1 8k → L by showing and then splitting the latter sum...
(2) Let (X, 2, ) be a measure space and let (.). be a sequence of...
(2) Let (X, 2, ) be a measure space and let (.). be a sequence of (E, BR)-measurable functions from X to R. Suppose that 'n pointwise and lim. S du SI du < 0. Prove that for all E E lim,- Jef du = sef du < 0.
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a...
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
1.Let Ai, A2,. . , Ak be collection of sets that partitions the sample space S....
1.Let Ai, A2,. . , Ak be collection of sets that partitions the sample space S. Which of the following are properties of the partition? B. Ain A2nnAk S. C. The sets A, A are mutually exclusive D. The sets A1, A2,... Ak are independent. E. Only A. and B. F. Only A. and C. G. Only A. and D. H. Only B. and C. I. Only B. and D. J. Only C. and D. K. Only A., B., and...
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x)...
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.)
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
3. Let Ei 2 E2 ? ... ? Ek ? ... be a decreasing sequence of...
3. Let Ei 2 E2 ? ... ? Ek ? ... be a decreasing sequence of nonempty, closed subsets of R”. (a) Prove that if Ej is compact, then n En +0. n=1 n= 1 (b) Find an example of a sequence as above with the property that n En = D. (Hint: In light of (a), none of the E; can be compact sets. Since each E; is closed by assumption, each of the E; must be unbounded. Also,...
6. In this problem, u is Lebesgue measure on R, while v is counting measure on N. Let =u X v be the product measure and let f: R *N → R be given by f(x, n) = 1 + (21 x)? Compute S f d). Remember that S itu du = arctan u + C.
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
(Exercise 4.13, reordered) Given a series ΣΧί ak, let 8,-Ση-i ak. Σχί ak is Cesaro summable if S1 + 82 +... +Sn lim n-+o converges. (a) Give an example of a series Σ00i ak that is Cesaro sum mable but not convergent (b) Prove that if 1 ak converges, then it is Cèsaro summable. Hint: Say the sequence of partial sums sn → L. Try to prove that =1 8k → L by showing and then splitting the latter sum...
(2) Let (X, 2, ) be a measure space and let (.). be a sequence of (E, BR)-measurable functions from X to R. Suppose that 'n pointwise and lim. S du SI du < 0. Prove that for all E E lim,- Jef du = sef du < 0.
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
1.Let Ai, A2,. . , Ak be collection of sets that partitions the sample space S. Which of the following are properties of the partition? B. Ain A2nnAk S. C. The sets A, A are mutually exclusive D. The sets A1, A2,... Ak are independent. E. Only A. and B. F. Only A. and C. G. Only A. and D. H. Only B. and C. I. Only B. and D. J. Only C. and D. K. Only A., B., and...
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.)
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
3. Let Ei 2 E2 ? ... ? Ek ? ... be a decreasing sequence of nonempty, closed subsets of R”. (a) Prove that if Ej is compact, then n En +0. n=1 n= 1 (b) Find an example of a sequence as above with the property that n En = D. (Hint: In light of (a), none of the E; can be compact sets. Since each E; is closed by assumption, each of the E; must be unbounded. Also,...
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