Let N be a nonempty set and B a o-ring on N.Let µ, v be two measures on B andH(A) < v(A),А В.Show thatfor all nonnegative measurable f : N → [0, 0∞].

Answered: Let N be a nonempty set and B a o-ring…

Let N be a nonempty set and B a o-ring on N. Let µ, v be two measures on B and μ(A) < ν(Α), A € B. Show that fdµ < / fdv or all nonnegative measurable f :N→ [0, ∞0].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 22E: Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].

See similar textbooks

Related questions

Q: Define an inner product space for continuous functions on [0, 1] as (S, g) = [f(x)g(x)ax with the…

Q: Let f be bounded and a be monotonically increasing in the closed interval a, b). If a' is Riemann…

A: Given: f is a bounded function on a,b. α is monotonically increasing on a,b. α' is Riemann…

Q: Let f(x) = x sin x. a. Compute f'(x). b. Show that for every M > 0 and every k > 1, there is xo > M…

Q: (b) Prove that if f,g are integrable on S, then so is f + g, and +9) = /. f + g. S S (First use part…

Q: Let ƒ be Riemann integrable on [a, b], and suppose g : [a, b] → R is a function such that g(x) =…

A: Given information f be a Riemann integrable on a,b g:a,b→R is a function such that fx=gx except for…

Q: Let f be bounded and a be monotonically increasing in the closed interval [a, b]. If a' is Riemann…

A: Given: f is a bounded function on a,b. α is monotonically increasing on a,b. α' is Riemann…

Q: Let S be a closed n-cube with v(S) > 0. Suppose g, h : S → R are two bounded #3 and integrable…

Q: Let {u. be a sequence of non-negative measurable functions and let f = Eu, almost everywhere on a…

A: The given sequence is f = ∑n-1∞un It is measurable at every point Now taking integration on both…

Q: Exercise 6. Let f be a function with Ifl<2 and C be the curve that connects 0 to 1 and then 1 to 1+i…

Q: Let S be the sawtooth function defined by if 0 <u <1 {:- и = (n)S и — 1 = {u-1 if1<us2 Prove that S…

Q: Prove that the family of operators T (t): L2 (0,1) → L2 (0,1), t > 0, where T (t) p (x) = 0 for 0 <r…

A: Let T'=Tt:L20,1→L20,1 be an operator for a fixed t≥0. To prove that this operator is a strongly…

Q: Given a measure space (X,U, u). Let {fn: ne N} and f be extended real-valued U-measurable functions…

Q: A function from Z ×Z to Z defined by f(m, n) =m-nis an onto function. True False

Q: Exercise 6. Let f be a function with Ifl4 O <1 None of these

Q: Suppose that f is a continuous function satisfying stæ) = a f(t)dt + * So f(t)dt + a %3D for all a,…

A: Solve the function

Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

Q: Let P denote the set of functions f in C([0,1)) with f(x) > 0 for all x. Then Pn B1(0) is…

A: Given that, P denote the set of function f in C0,1

Q: A bounded function f: A R is integrable if and only if for every e > 0 there is a partition P of A…

Q: Let f,(x) = x E R. Prove that f 0 uniformly on R. COs nr %3D

Q: Let f(z) := z5 . Show that f is invertible in a neighbourhood of i and calculate (f-1)'(f(i)).

Q: Let f (x) be a nonconstant element in Z[x]. Prove that k f (x)l is notmaximal in Z[x].

Q: Suppose that ƒ is continuous and F(x) = g> 0, g'>0 and integrable on [a, b) and that lim g(x) = I-b-…

Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

Q: Let N C C be a domain (that is, a connected open subset), and let f: N → C be holomorphic. Prove…

Q: Let f be a bounded function on [a,b], and suppose that (PnnEN is a family of partitions on [a,b]…

A: Given that f is a bounded function on [a,b]. (Pn)n∈ℕ is a family of partitions on [a,b] Assume that…

Q: Prove that the family of operators T (t): L2 (0,1) → L2 (0,1), t > 0, where T (t) p (x) = 0 for 0 <r…

A: Fix an operator Tt:L20,1→L20,1, for a fixed t. To prove that Tt is a strongly continuous semigroup.…

Q: Let f be a bounded function on [a,b] and suppose that (P)aEN is a family of partitions on [a,b]such…

A: Given: Let f be a bounded function on [a,b] and (Pn)n∈N is a family of partitions on [a,b] such that…

Q: 3. Use the Bolzano Weierstrass Theorem to prove that if f is a continuous function on [a, b), then f…

Q: Let f be defined and continuous on [a,b]. Then for each x E [a,b] there exists a polynomal p(x) such…

Q: A set A C [0,1] is dense in [0, 1] iff every open interval that intersects [0, 1] contains a point…

Q: Let f ∈ L (U, V ), g ∈ L (V, W) and let U, V be finite-dimensional. Prove that(a) dim im(g ◦ f) =…

A: Note: Kindly re-post the question for the proofs of "dim im(g ◦ f) ≤ min(dim im f, dim im g);" and…

Q: Let E = {p1, P2,, PN} be a finite subset of a closed interval [a, b), and g : [a, b] →R be a…

Q: Suppose f,9 : D R are continuous on D. uniformly continuous on D. Prove that f +g is uniformly

Q: Let f be a continuous function on [a, b] such that f(r)a" dx = 0 for every n > 0. Prove that f = 0…

Q: Find all volves of o,b ond c such that the func Ef×メ goo)= てメー1) 1チxg? 21 if x= 1 is continuous at…

A: Given, gx=ax3+bx2+cx+1x-12, if x≠12, if x=1

Q: 8. If f is conformal on a domain D which is symmetric with respect to the real axis, is f(z)…

A: Given that f is conformal on a domain D which is symmetric with respect to the real axis.

Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

Q: (2) (a) Consider a function f, defined in some open neighbourhood of ro ER, i.e., f : (a; b) → R, a…

Q: Suppose that f is integrable on [a,b], and that g(x) is defined for all x E[a,b] and ,exists and…

A: Given: To explain the given statement as follows, To prove that exists and equals .

Q: 7. Let (S, p) be a metric space and T : S → S be a contraction mapping. Prove that T is uniformly…

Q: Let A = [0, 1] × [0, 1] C R². Suppose f : A → R is continuous on A (re (a) suprɛ[0,1] f(x, y). Prove…

A: (a) Given, A =0, 1×0, 1⊆ℝ2. Suppose that f:A→ℝ is continuous on A(relative to A). To prove…

Q: A set of real numbers D is called dense if and only if every open interval contains at least one…

Q: f(z) = Im z is continuous ylgn O ihi O

A: Use the property of continyity in C which is if real and imaginary part of functiom is continuous in…

Q: Let f be bounded and a be monotonically increasing in the closed interval [a, b]. If a' is Riemann…

Q: Exercise 6. Letf be a function with Ifl<2 and C be the curve that connects 0 to 1 and then 1 to 1+i.…

Q: Exercise 6. Let f be a function with Ifl<2 and C be the curve that connects 0 to 1 and then 1 to 1+i…

Q: Let f be defined and continuous on [a,b]. Then for each x E[a,b] there exists a polynomal p(x) such…

Q: Consider the function f: Z x Z → Z defined by f (m, n) = 2m – 2n for all (m, n) E Z × Z. (a) Is f…

A: A function f: A→B is said to be one-one if f(x1)=f(x2) ∀x1, x2∈A ⇒x1=x2 In words,…

Question

Let N be a nonempty set and B a o-ring on N.
Let µ, v be two measures on B and
H(A) < v(A),
А В.
Show that
for all nonnegative measurable f : N → [0, 0∞].

Expert Solution

Step 1

Let $Ω$ be a non- empty set.

And $B$ a sigma ring on $Ω$ .

Let $μ, ν$ be the two measures on $B$

And $Ω (A) \leq ν (A)$ , $A \in B$ .

The objective is to show that $\int f d μ \leq \int f d ν$ for all non-negative measurable $f : Ω \to [0, \infty]$ .

Step by step

Solved in 2 steps

Knowledge Booster

$Relations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

28. Let where and are nonempty. Prove that has the property that for every subset of if and only if is onto. (Compare with Exercise 15c.) Exercise 15c. c. For this same and show that.
If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]
2. Prove the following statements for arbitrary elements of an ordered integral domain . a. If and then . b. If and then . c. If then . d. If in then for every positive integer . e. If and then . f. If and then .
Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd
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. 3. Let where A and B are nonempty. Then for every subset S of A.