Question

In: Advanced Math

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that...

Let ? be a ?−algebra in ? and ?:? ⟶[0,∞] a measure on ?. Show that ?(?∪?) = ?(?)+?(?)−?(?∩?) where ?,? ∈?.

Expert Solution

Proof:

Let be a algebra in and a measure on .

Given that and are measurable sets.

If either or have infinite measure then the inequality trivially hold.

Assume that and have finite measure.

Write as disjoint union.

By finite additivity and excision property we get

Colby Messinger answered 2 years ago

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following...

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following sets. (a) S1 = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)} (b) S2 = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0. (b) Show that A is symmetric if and only if A^2= AA^T

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative...

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative group {1, −1}.

Let A be such that its only right ideals are {¯0} (neutral element) and A. Show...

Let A be such that its only right ideals are {¯0} (neutral element) and A. Show that A or is a ring with division or A is a ring with a prime number of elements in which a · b = 0 for any, b ∈ A.

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A)...

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A) > 0 then A is positive definite. (trace of a matrix is sum of all diagonal entires.)

Let A be a non-negative random variable (A>0) a) If A is discrete, show that E[A]...

Let A be a non-negative random variable (A>0) a) If A is discrete, show that E[A] >= 0 b) If A is continous , show E[X]>=0

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the...

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the following properties hold: (a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x). (b) If g(x)|f(x), then g(x)h(x)|f(x)h(x). (c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x). (d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F \ {0}

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show...

Let f be a differentiable function on the interval [0, 2π] with derivative f' . Show that there exists a point c ∈ (0, 2π) such that cos(c)f(c) + sin(c)f'(c) = 2 sin(c).

Let {λn} be a sequence of scalars that converges to zero, limn→∞ λn = 0. Show...

Let {λn} be a sequence of scalars that converges to zero, limn→∞ λn = 0. Show that the operator A : ℓ2 → ℓ2 , A(x1, x2, ..., xn, ...) = (λ1x1, λ2x2, ..., λnxn, ...) is compact. What is the spectrum of this operator?