Suppose that &#x03BD;<!-- ν --> &lt;&lt; &#x03BC;<!-- μ --> . Then we can find a non-negative

antennense 2022-07-09 Answered
Suppose that ν << μ. Then we can find a non-negative f s.t.
ν ( E ) = E d ν = E f d μ
So far, things seem clear to me. My question is the following: Though it makes intuitive sense, how can we be sure that
E d ν = E f d μ E g d ν = E g f d μ
for all integrable functions g? In the top answer to the question I linked to above, the following claim is made:
For every integrable g, the following formula holds:
E g d ( E f d μ ) = E g f d μ
It therefore seems that a justification/proof of this claim would answer my question.
I have been exposed to some measure theory and integration theory a few years back, and as I was revising some material recently, this claim was not clear to me. Perhaps this claim is obvious, and my confusion simply arises from a poor understanding of important definitions. Either way, any help in understanding this claim is much appreciated.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Jordan Mcpherson
Answered 2022-07-10 Author has 16 answers
The result is given as a theorem in the book Real Analysis: Measures, Integrals and Applications, by Makarov and Podkorytov (p. 146). The proof in that book is essentially the same as the argument given by Prahlad Vaidyanathan.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-14
Who knows
If I define algebra F ( A ) generated by A, collection of subsets of S (the universal set) as the intersection of F , algebra superset of A:
F ( A ) = a l g e b r a   F A F
What if A is an infinite (either countable or uncountable) set? Algebra, unlike σ-algebra, guarantees being closed under finite Boolean operations. Here, finite(in the definition of algebra) and infinite(in the setting) confuses me. e.g. A is the collection of intervals ( , x ]( x R ) and S = R , what F ( A ) be like? Any help will be appreciated!
asked 2022-06-03
I have an exercise in my measure and integration theory course and I'm a little bit stuck with the σ-rings and σ-algebras. The exercise goes as follows:
Let X be a set and ε = { { x } | x X }. Determine σ ( ε ), the smallest σ-algebra that contains epsilon.
I know that ε = { { x } | x X } is the set that contains the singeltons as elements.
My first problem when defining the σ-algebra is whether it has to contain the empty set AND the set X, OR the empty set and the set ε? Because this is one of three conditions for forming a σ-algebra.
The second condition, that σ ( ε ) has to contain the complement of a subset A σ ( ε ), is (I think) clear for me. But then the third condition, that it has to contain the the union of all subsets of σ ( ε ), is also unclear. I am not sure how to "build" this union.
asked 2022-06-06
Measurements of some variable X were made at an interval of 1 minute from 10 A.M. to 10:20 A.M.. The data, thus, obtained is as follows:
X:60,62,65,64,63,61,66,65,70,68,63,62,64,69,65,64,66,67,66,64,50.
The value of X which is exceeded 50% of the time in the duration of measurement, is
(i) 69
(ii) 68
(iii) 67
(iv) 66

Answer:
It is a multiple choice question and it is general knowledge question.
The answer is 68.
But I am unable to understand the question. What exactly says the question?
asked 2022-06-14
Consider a sequence of identically distributed random variables ( X n ), with E | X n | finite. Define Y n = sup ( | X 1 | , , | X n | ) / n. I must prove that Y n converges to 0 in L 1 .
If Ω is the domain of the random variables, then n Y n ( ω ) x if, and only if, | X i | ( ω ) x / n for i = 1 , , n. Thus, Y n 1 ( ( , x ] ) = i = 1 n X i 1 ( ( , x / n ] ). Then P ( Y n x ) = P ( i = 1 n X i 1 ( ( , x / n ] ) ). Does it follow from this that Y n is identically distributed to some of the | X i | ? Are there any hypothesis that may be missing?
Edit: there was a typo in the original question
asked 2022-05-22
I have a random variable ξ with function distribution (f.d.) G . Define the p −dimensional random vector as
X = ( ψ 1 ξ , . . . , ψ p ξ ) F ( F  is f.d. )

Given a > 0, consider A = [ a , a ] p . I want to find an expression for:
A d F
involving G and the ψ s
My attempt is first try to find the f.d. of X:
F ( x 1 , . . . , x p ) = P [ ψ 1 ξ x 1 , . . . , ψ p ξ x p ] = P [ ξ x 1 ψ 1 , . . . , ξ x p ψ p ] = P [ ξ n min { x 1 ψ 1 , . . . , x p ψ p } ] = G ( min { x 1 ψ 1 , . . . , x p ψ p } )
Even with these expressions in hand, I'm having serious trouble finding an expression for A d F.
asked 2022-06-16
Let n N, ( X , A , μ ) be a measure space and E 1 , . . . , E n A. For each j { 1 , . . . , n } define
C j ∶= { x X | x E k for exactly  j  indices , k { 1. . . , n } } .
I want to find a general expression for all the C j since I need to prove that each C j is a measurable set, I made an example and it is as follows
With n = 3
C 1 = ( E 1 E 2 c E 3 c ) ( E 1 c E 2 E 3 c ) ( E 1 c E 2 c E 3 )
C 2 = ( E 1 E 2 E 3 c ) ( E 1 E 2 c E 3 ) ( E 1 c E 2 c E 3 )
C 3 = k = 1 3 E k
So for C n we already have a general expression:
C n = k = 1 n E k
for C 1 :
C 1 = j = 1 n ( E j k j n E k c )
I was thinking of introducing permutations but actually I don't know if it works, someone can guide me a bit to express the C j sets in a manageable way
asked 2022-05-27
I'm attempting to understand some of the characteristics of Posiitive Operator Value Measurement (POVM). For instance in Nielsen and Chuang, they obtain a set of measurement operators { E m } for states | ψ 1 = | 0 , | ψ 2 = ( | 0 + | 1 ) / 2 . The end up obtaining the following set of operators:
E 1 2 1 + 2 | 1 1 | , E 2 2 1 + 2 ( | 0 | 1 ) ( 0 | 1 | ) 2 , E 3 I E 1 E 2
Basically, I'm oblivious to how they were able to obtain these. I thought that perhaps they found E 1 by utilizing the formula:
E 1 = I | ψ 2 ψ 2 | 1 + | ψ 1 | ψ 2 |
However, when working it out, I do not obtain the same result. I'm sure it's something dumb and obvious I'm missing here. Any help on this would be very much appreciated.