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. For z eR, let 6 be the set function defined by Describe the class of 5-measurable sets (i.e. se...
Th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) ...
th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) Prove or find a counter example for the statement "a somewhat continuous func- (52) Let EC Rm, f: ER, and a e E. Prove or disprove that f is continuous at a there is a 6 0 such that whenever z,y D and la -vl <6, then f()-)e es tion is continuous." if and only if given any...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, :...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
Please be detailed in your answer. Thank you. 1. Let f g be measurable functions defined...
Please be detailed in your answer. Thank you.
1. Let f g be measurable functions defined on a measurable domain E. Let A, = {x € Elg(x) = 0}. It is clear that the domain of() is A . Prove the following: a. A, is a measurable set. b. (1) is a measurable function on Ap. Hint: Show that for every a € R, {xea |^)(x) < a} is m easureable. Start by proving that {x e Aol (6) (x)...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
(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...
(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...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are
2. Consider the relation E on Z defined by...
Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3...
Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3] Z). A relation R is defined on the set Q+ of positive rational numbers by R b if
Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3] Z). A relation R is defined on the set Q+ of positive rational numbers by R...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n +...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
th 5. (14pt) A function defined on D C R is said to be somewhat continuous if for any e >1, (51) Prove or find a counter example for the statement "a somewhat continuous func- (52) Let EC Rm, f: ER, and a e E. Prove or disprove that f is continuous at a there is a 6 0 such that whenever z,y D and la -vl <6, then f()-)e es tion is continuous." if and only if given any...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
Please be detailed in your answer. Thank you.
1. Let f g be measurable functions defined on a measurable domain E. Let A, = {x € Elg(x) = 0}. It is clear that the domain of() is A . Prove the following: a. A, is a measurable set. b. (1) is a measurable function on Ap. Hint: Show that for every a € R, {xea |^)(x) < a} is m easureable. Start by proving that {x e Aol (6) (x)...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
(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...
(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...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are
2. Consider the relation E on Z defined by...
Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3] Z). A relation R is defined on the set Q+ of positive rational numbers by R b if
Let H-{2m : m ajbe H. (a) Show that R is an equivalence relation. (b) Describe the elements in the equivalence class [3] Z). A relation R is defined on the set Q+ of positive rational numbers by R...
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
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