Questions tagged [product-measure]

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1answer
62 views

Fibered product of probability spaces

Let $(\Omega_1, \mathcal{F}_1, \mu_1)$, $(\Omega_2, \mathcal{F}_2, \mu_2)$ and $(T, \mathcal{G}, \nu)$ be probability spaces. A probability space morphism is a measurable function that preserves the ...
3
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1answer
61 views

Show the left shift $T$ and $T^{-1} $ are measurable.

Let $X=\{0,1 \}^{\mathbb{Z}}$. So an element of $X$ is given by a sequence $(x_i)_{i\in \mathbb{Z}}$, where $x_i \in \{0,1 \}$. Let $\mu_i$ be a probability measure on $(\{0,1 \}, \mathcal{P}(\{0,1 \})...
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18 views

Product measure, orthogonal subspaces

Let $X$ be a Hilbert space with scalar product $(\cdot, \cdot)$ and with an orthonormal basis $B=\{\phi_i\}_{i\in \mathbb N}$. Morevoer, let $N$ be a positive integer and let $P^N$ be the projection ...
2
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1answer
42 views

Locally finite Borel measure on $\mathbb{R}^2$ that is not a product measure.

In the construction of Lebesgue-Stieltjes measures on $\mathbb{R}$, I have learned that a Borel measure that is finite on bounded intervals corresponds to a right-continuous increasing real-valued ...
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1answer
53 views

Atomless?: product measure of an atomless measure and a measure which has atoms

I wonder whether the product measure of an atomless measure and a measure which has some atoms is atomless. Let us recall that given a measure space $(X, \mathscr A, \mu)$, the set $A \in \mathscr A$ ...
1
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1answer
39 views

Show that $f$ is measurable with respect to the product measure $c \times c$.

I am really stuck on this question. I am new in real analysis but interested. yet, I am not familiar with this. Let $(X, \mathcal{A},\mu) = (Y, \mathcal{B}, v) = (\mathbb{N}, \mathcal{M}, c)$. Here, ...
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0answers
18 views

Prove $m^*_{p+q}=(m_p\otimes m_q)^*$

Setting Semi-ring $P_1=\{\prod_{i=1}^{p+q} [a_i,b_i):a_i\leq b_i,\bar{a},\bar{b}\in\mathbb{R}^{p+q}\}$ And the measure to be on the semi-ring $m_{p+q}([\bar{a},\bar{b})=\prod_{i=1}^{p+q}(b_i-a_i)$ and ...
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0answers
33 views

How do you show that the function $f(x)=\nu(P_x)$ is measurable?

There is a pretty classic question that is: Let $X=Y=[0,1]$, $\mathcal{M}=\mathscr{B}_{[0,1]}$, $\mathcal{N}=2^{[0,1]}$, let $\mu$ be Lebesgue measure on $\mathcal{M}$ and let $\nu$ be counting ...
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1answer
65 views

Show there exists a set $E \subset \mathbb{R}^2$ such that the cross sections are open subsets but $E \notin \mathcal{B}_2$.

Show there exists a set $E \subset \mathbb{R}^2$ such that the cross sections $[E]_a$ and $[E]^a$ are open subsets for every $a \in \mathbb{R}$ but $E \notin \mathcal{B}_2$. This is a question from ...
3
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1answer
67 views

$f: X \rightarrow \mathbb{R}$ is measurable iff $\{(x, t): t=f(x)\} \in \mathcal{F} \otimes \mathcal{B}(\mathbb{R})$.

A function $f: X \rightarrow \mathbb{R}$ is measurable if and only if its graph is measurable, $\{(x, t): t=f(x)\} \in \mathcal{F} \otimes \mathcal{B}(\mathbb{R})$. This is easy when we consider the ...
0
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1answer
47 views

$K=\{E|f^{-1}(E)\in \mathcal{S} \}$ is a $\sigma$ algebra [closed]

This is a problem I have seen while self studying measure theory: Let $(X,\mathcal{S})$ be a measure space, and $E\in \mathcal{S} \otimes \mathcal{S}$, with $f(x) = (x,x)$ then $K=\{E|f^{-1}(E)\in \...
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44 views

Getting Borel-measurability from a product measure space

I have that the graph of a function $G(f) = \{ (x, f(x)) : x \in \mathbb{R} \} $ for $f : \mathbb{R} → \mathbb{R} $ is such that $G(f) ∈ B \otimes B$. Here $B$ is the $\sigma$-algebra of Borel sets. ...
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2answers
48 views

Finding $E \in \mathcal A \otimes \mathcal B$ such that $E \neq E^y \times E_x,$ for some $x \in X, y \in Y.$

Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be two measure spaces. What has been stated in my book is that $\mathcal A \times \mathcal B$ may not necessarily be a $\sigma$-algebra of subsets ...
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0answers
51 views

A question of why these integrands can be taken out of the integrals.

Rudin says that (2) can be rewritten as (4), as seen below. However, both integrands in (2) respectively depend on $x$ and $y$, which implies that (2) could not be rewritten as (4). Is there any ...
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2answers
62 views

Random variable greater than another random variable - measure theoretic argument?

Suppose $\xi,\eta$ are two independent, identical random variables that are non-degenerate. I want to show that $\mathbb{P}(\xi<\eta) = \mathbb{P}(\eta<\xi) > 0$ through a measure-theoretic ...
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0answers
23 views

Show that there exists a unique measure $\pi$ on $\mathcal{B}[0,\infty)\otimes\mathcal{P}(\mathbb{N})$

I have already shown that $\Gamma\in \mathcal{B}[0,\infty)\otimes\mathcal{P}(\mathbb{N})$ iff $\Gamma = \bigcup_{j\in\mathbb{N}} A_j\times \{j\}$ where $A_j \in\mathcal{B}[0,\infty)$ for all $j$. I am ...
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0answers
61 views

Confusions with Fubini’s Theorem

Fubini's Theorem: Let $\mu,\nu$ be two $\sigma$-finite measures and $f$ be a measurable function. If $f \ge 0$ or $f \in L^1(\mu\times \nu)$, then $$ \int f ~d \mu \times \nu = \iint f ~d\mu ~d\nu = \...
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0answers
42 views

Area under a curve proof using product measure

Let $u:\mathbb{R}\to[0,\infty]$ be a Borel measurable function, define the set $S[u] = \{(x,y):0\leq y\leq u(x)\}$ and let $\lambda^n$ be the Lebesgue measure for $\mathbb{R}^n$. I am trying to show ...
1
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1answer
42 views

Extending the definition of stochastic integral from simple processes

I am reading stochastic integration from Brownian Motion And Stochastic Calculus by Karatzas and Shreve. In the course of extending the definition of the stochastic integral from simple processes to ...
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1answer
171 views

Product measure of Lebesgue and counting measure on discrete and usual topology on $\mathbb{R}$

$\textbf{Corollary}$: Let $X,Y$ be locally compact Hausdorff spaces. Let $\mu$ and $\nu$ be regular Borel measures on $X$ and $Y$ respectively. If $E$ is a Borel subset of $X\times Y$ that is included ...
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1answer
47 views

Determine if $A$ and $B$ are independent (Product Measure, Independence)

Consider $\Omega_1 = \mathbb{N}_0$, $\Omega_2 = \mathbb{R}$, $\mathbb{P}_1 = Poi(5)$ and $\mathbb{P}_2 = Normal(1,2)$. On $\Omega = \Omega_1 \times \Omega_2$ we use the the product measure $\mathbb{P}$...
1
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1answer
64 views

An application of Fubini's theorem I don't understand

Here is a result asserting that, under some technical conditions, we can replace a Borel almost everywhere homomorphism by a Borel homomorphism. It is taken from Zimmer's book "Ergodic Theory and ...
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1answer
106 views

Measure Theory: $X$, $Y$ independent, proof that $P_{X+Y} = P_{X} * P_{Y}$

I am studying measure theory, and I bumped in the the following: Let $X$ and $Y$ be two independent random variables with distributions $P_X$ and $P_Y$ respectively. A lot of sources conclude (or use ...
1
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1answer
69 views

Which elements are contained in an infinite product space?

I have a question about the formal treatment of infinite product measures in probability calculus. Take for example the model of an infinite coin toss. If $(\Omega_i,\mathcal A_i,P_i)$ is the ...
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2answers
42 views

Joint Random Variables

I am trying to clarify the meaning of a joint pdf. A collection of random variables $\{X_i\}_{i=1}^{n}$ that are i.i.d. defined on the space $(\Omega, F, P)$, we define their joint probability as: ...
1
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1answer
38 views

Equality of certain integrals over product space

Let $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ be measurable spaces. (I'm happy to restrict this to standard Borel spaces.) Suppose there are finite measures $\mu$ and $\nu$ on $X$ and a finite kernel $k$ ...
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1answer
80 views

If $\lambda \perp \mu$ and $\rho \perp \nu$, then is it true that $\lambda \times \rho \perp \mu \times \nu$?

Let $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$ be measure spaces. Let $\lambda$ and $\rho$ be signed measures on $\mathcal{M}$ and $\mathcal{N}$ respectively. Suppose $\lambda \perp \mu$ ...
2
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1answer
74 views

Tonelli's theorem holds for an arbitray $(Y,\mathcal{Y},\nu)$ measurable space in this case

My question is the exercise 10.M) of Bartle's book. I want to prove that Tonelli's theorem holds for an arbitrary $(Y,\mathcal{Y},\nu)$ measurable space if $(X,\mathcal{X},\mu)$ is the measurable ...
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0answers
76 views

Computing product measure.

I'm dealing with this case: Let $(X,\mathcal{X},\mu), (Y,\mathcal{Y},\nu)$ measurable spaces, with $X=Y=[0,1],\mathcal{X}=\mathcal{Y}=$ Borel in $[0,1].$ Let $\mu$ the Lebesgue Measure and $\nu$ the ...
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0answers
69 views

Integrability in product space

Let $(X=\Bbb{N},\mathcal{X}=\mathcal{P}(\Bbb{N}),\mu)$ a measurable function, where $\mu$ is the counting measure. Let $(Y,\mathcal{Y},\nu)$ an arbitrary measurable space. First, I already proved ...
2
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1answer
384 views

Prove graph of measurable function is measurable in $\sigma$-finite case and that the product measure is $0$.

Let $(X, \mathcal{A}, \mu)$ be a $\sigma$-finite measure space, and let $f: X \to \mathbb{R}$ be measurable. Then, $\Gamma(f)$, the graph of $f$ defined as $$\Gamma = \{(x,y) \in X \times \mathbb{R}: ...
3
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1answer
307 views

Approximate a measurable function by simple functions in a product measure space

This is a theorem in Yeh's Real Analysis : Let $(X, \mathbf{A}, \mu)$ and $(Y, \mathbf{B}, v)$ be two finite measure spaces. Consider the product measure space $(X \times Y, \sigma(\mathbf{\mathbf { ...
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0answers
61 views

Product measure without integration

Is it possible to prove the existence of the product measure without the concept of integration? My thoughts are focused on trying to prove it indirectly using the Carathéodory's extension theorem. ...
2
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1answer
214 views

Measures absolutely continuous wrt Dirac measure

Suppose we have a probability space $(X,\mathcal{F},\mu)$ and $x_0\in X$. Consider the measure $\mu\times \delta_{x_0}$, where $\delta_{x_0}$ is the Dirac measure at $x_0$. Then is it true that the ...
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0answers
50 views

$E \subset [0, 1]^{2}$ s.t. $m(E_{x}) = 1$ a.e. $x \in [0, 1]$. Show $m(E_{y}) = 1$ a.e. $y \in [0, 1]$.

Let $E \subset [0, 1]^{2}$ be measurable and let $E_{x} := \{y \in [0, 1] : (x, y) \in E\}$, $E_{y} := \{x \in [0, 1] : (x, y) \in E\}$. If $m(E_{x}) = 1$ a.e. $x \in [0, 1]$ then show that $m(E_{y}) =...
2
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1answer
132 views

Rick Durrett, Probabilty Theory and Examples, Lemma 2.2.8

I have a question from Rick Durrett's Probability Theory and Examples. In that book, Lemma 2.2.8 If $Y \ge 0$ and $p > 0$, then $E(Y^p) = \int^{\infty}_{0} py^{p-1}P(Y>y)\,dy $ Proof. Using ...
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1answer
35 views

A Question about Product Measures

I'm working on the following problem: Let $\mathcal{M}$ be a $\sigma$-algebra on subsets of $X$ and $\mathcal{N}$ be a $\sigma$-algebra on $Y$. Discuss the validity of the following statement: $A \...
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1answer
35 views

not every measure on a product space has a product measure [closed]

Define $f: [0, 2\pi) \to \mathbb{R}^2$ by $f(t) = (\cos (t), \sin(t))$. So, $f([0, 2\pi)) = C$ is a circle. Define a measure $\rho: \mathcal{B}(\mathbb{R}^2) \to [0, \infty]$ by $\rho(E) = \lambda(f^{-...
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0answers
250 views

Extending Fubini's Theorem to general joint distributions

Take Fubini's theorem as stated above. Say there exists $(M,\mathscr{M}, \mu)$ and $(N,\mathscr{N},\nu)$, then the product measure is such that $\pi(A\cap B)=\mu(A)\nu(B)$ where $\pi$ is defined over ...
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2answers
139 views

Calculating a product measure with Fubini-Tonelli's theorem

I am kind of lost on the following problem. Let $${f : (X, \mathcal{A}, \mu) \rightarrow (\mathbb{R_{\geq}}, \mathcal{B}(\mathbb{R_{\geq}}), \lambda)}$$ with ${\lambda}$ being the Lebesgue-measure. 1)...
0
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1answer
43 views

Calculate product measure of off-diagonal set

Let $(E,\mathcal E)$ be a measurable space $\mu_i$ be a probability measure on $(E,\mathcal E)$ $\lambda:=\mu_1+\mu_2$ and $$f_i:=\frac{{\rm d}\mu_i}{{\rm d}\lambda}$$ $Q$ denote the measure with ...
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0answers
46 views

Computing iterated integrals over positive integers (question about stack exchange solution)

I am referring to this question: here How does one go about computing the iterated integrals? I am confused as to how the first one simplifies to $f(1,1)$ and why the second one has: $\sum_{y} 2^{-...
1
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1answer
144 views

Every section null implies null in product measure

Let $A\subset[0,1]^2$ be a set such that every section $A_x=\{y:(x,y)\in A\}$ is a null set in $[0,1]$. Can we conclude that $A$ is a null set in $[0,1]^2$? Some context: It is a standard fact that ...
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0answers
38 views

Is $Tf$ $\mu$- measurable?

Suppose that $X$ and $Y$ are sets with measures $\mu$ and $\nu$ defined on them. Suppose $K(x, y)$ is $\mu × \nu$ measurable and that there is a constant $C > 0$ such that $\int_{X}|K(x, y)| dµ(x) ...
4
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1answer
92 views

Strong convergence in $L^1$, $\mathbb{R}^d$ with $d>1$

A sequence of positive function $f_n(x)\rightarrow f(x)$ pointwise on the unit ball $B:=\{x:|x|\leq 1\}\subset \mathbb{R}^d$ as $n\rightarrow \infty$. Suppose the sets $\Lambda_t^n=\{x:f_n(x)\geq t\}$ ...
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0answers
440 views

The collection of measurable rectangles form an algebra.

Question: Show that the collection of measurable rectangles form an algebra. Denote $\Sigma$ by $\sigma$-algebra. I found the following set operations. Let $A_1, A_2 \in \Sigma_A$, and $B_1, B_2 \...
1
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1answer
37 views

Sectioning sets in product measures

I'm reading through Capinski and Kopp's Measure, Integral and Probability and stumbled across Theorem 6.4 about sectioning sets in product measures: If $A$ is in the product $\sigma$-field $\...
3
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2answers
169 views

Prove $\{(\omega,y) \in \Omega \times [0,\infty) \ | \ y \leqslant f(\omega) \}$ is in the product $\sigma$-algebra $\mathcal{A} \otimes \mathcal{B}$

We have a $\sigma$-finite measure space $(\Omega,\mathcal{A},\mu)$, a measurable nonnegative function $f:\Omega \to \mathbb{R}$ and \begin{equation} G_f := \{(\omega,y) \in \Omega \times [0,\...
1
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1answer
151 views

double integral of f zero over subsets of $[0,1]^2$ implies $f=0$ almost everywhere

Here's the statement of the problem: Let f $\epsilon L^1([0,1]^2,\mathscr{B}(\mathbb{[0,1]^2}),m)$, where $\mathscr{B}(\mathbb{[0,1]^2})$ is the Borel $\sigma$-algebra and m is the two-dimensional ...
5
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1answer
427 views

Show a function defined by an integral is Borel measurable.

Let $f: \left [ 0, 1 \right ] \times \left [ 0, 1 \right ] \rightarrow \mathbb{R}$ satisfy: $f_x\left ( y \right ):= f\left ( x, y \right ) : \left [ 0, 1 \right ]\rightarrow \mathbb{R} $ is Riemann ...