For questions about the structure of product space, in the context of topology or measure theory for example. Use other tags to indicate the context.

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2
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17 views

Show that iterated integral of indicator function over product space defines a countably additive (product) measure.

Let $\Sigma = \Sigma _2 \times \Sigma _2 $ Given that $x $-and $y $ sections are $\Sigma _1 $ - and $\Sigma _2 $ measurable respectively, and that for $F \in \Sigma $, $\int ( \int I _F \, d \mu _1 ...
2
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1answer
35 views

Product measures and $\sigma-$ finite measures

Problem similar to folland chapter 2 problem 51. The actual problem in Folland mentions that $X,Y$ are not necessarily $\sigma-$finite. Then how can I use Fubini-Tonelli theorem?
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0answers
27 views

Product measures in measure theory

Suppose $(X,M,\mu)$ and $(Y,N,v)$ are $\sigma-$finite measure spaces. Is the following calculation correct? $\int \chi_E(x,y) d\mu=\int (\chi_E)^y(x) d\mu=\int \chi_{E^y}(x) d\mu=\mu(E^y)$ for a ...
3
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2answers
60 views

Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the product topology on $\mathbb{R}^m \times \mathbb{R}^n$

I'm attempting to teach myself topology for graduate school this summer, but I'm having a tough time. I'm trying to prove that the Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the ...
2
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2answers
177 views

Infinite product of probability measures is a premeasure

This is an exercise from Real Analysis by Stein and Shakarchi (Chapter 6, Exercise 15). Given infinitely many measure spaces $(X_i, \mathcal M_i, m_i)$, each of which has measure 1, one can define an ...
2
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2answers
108 views

The topology a line inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$, or of $\mathbb{R}_l \times \mathbb{R}_l$ (Munkres)

I have a question about Exercise 8, Chapter 2, Pag. 92, from the Munkres's book: If $L$ is a straight line in the plane, describe the topology $L$ inherits as a subspace of $\mathbb{R}_l \times ...
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56 views

Lebesgue-integrable and existence of integral

I have given the following function $$ f(x,y) = \begin{cases} 1 &, \ x \in \mathbb{Q} \\ 2y & , \text{ otherwise} \end{cases} $$ This is a measurable function in sense of Lebesgue. Now, I ...
3
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1answer
121 views

Questions on Fubini's Theorem and $\sigma$-finite measure?

I asked a question about this a several days ago, but I think I have a better formulated question now. The reason I did not just edit the last question about this is that I feel the answers I got ...
1
vote
1answer
26 views

The general form of a measurable set in a product measurable space

Hi everyone: Suppose that $(X,\mathfrak{M},\mu)$ and $(Y,\mathfrak{N},\nu)$ are two measure spaces and consider the product measure space $(X\times ...
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48 views

Proof: Product Topology Question XxY

If $f$ is maps from topogical spacce $Z$ to $X\times Y$ so: $f$ is continuous iff : $\begin{cases} (p_X)\circ f: Z \rightarrow X\times Y \rightarrow X \\ (p_Y)\circ f: Z \rightarrow X\times Y ...
4
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1answer
75 views

Product of T1 spaces is T1

I am trying to prove that the product of T1 spaces is also T1. Here is a proof, is it correct? $\{ X_i \}_{i \in I}$ are T1 $\Rightarrow$ $\prod_{i \in I} X_i$ is T1 Proof: Let $\bar{x} = ( ...
0
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1answer
16 views

continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous

Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$. If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$ $Y_1 \times Y_2$ is a topological space with ...
0
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1answer
30 views

Prove $ \varphi ( x ) = \lambda ( Q_x ) $ is $ \mathcal{X} $-measurable

I'm reading Thoerem 8.6, page 163, Chapter 8 from "Real and Complex Analysis (Third Edition)" by Walter Rudin; it seems that Rudin never proves that $\lambda(Q_x)$ (defined below) is a ...
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0answers
66 views

Probability that a given countable sequence is contained in a random countable sequence

Let a probability space $\langle \Omega, F, Pr \rangle$ be given. Consider the countable product space $\langle \Omega^*, F^*, Pr^* \rangle$ (that is, $\Omega^*$ is the set of countable sequences of ...
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1answer
123 views

A generalization of the generalized tube lemma

I am trying to prove the following generalization of the generalized tube lemma: Let $\{X_t\}_{t \in T}$ be a family of Hausdorff spaces and $\prod_{t \in T}A_t$ be a compact subset of $X=\prod_{t ...
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1answer
140 views

What is the definition of a product topology?

I am new to advanced mathematics and I recently started reading a book on topology. I am struggling to understand what it is saying in this paragraph. This is what it says: Let $E_i$ ...
3
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2answers
117 views

Product topology on $X \times Y$ the smallest topology when $f(x, y) = x$ and $g(x, y) = y$ are continuous functions?

$X$ and $Y$ are topological spaces and let $f : X \times Y \to X$ and $g : X \times Y \to Y$ be maps such that $f(x, y) = x$ and $g(x, y) = y \ \forall (x, y) \in X \times Y$ . Show that the product ...
0
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1answer
46 views

Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$

Full question: A function $f$ from a space $X, \tau$ to a space $Y, \tau'$ is said to be open if $f(V)$ is open in $Y$ whenever $V$ is an open subset of $X$. Prove that the projection $p_i$ from the ...
2
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2answers
38 views

Show that is not a sigma algebra

I would like to pose a question to you, and I will appreciate any hint or partial solution for my question. Let $\left(\Omega, \mathcal{F}\right)$ be a measurable space, and let $\left(\Omega \times ...
2
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0answers
55 views

Is $\sigma$-finiteness necessary for the existence and uniqueness of product measure?

Let $(X,\mathfrak{B}_X,\mu_X)$ and $(Y,\mathfrak{B}_Y,\mu_Y)$ be $\sigma$-finite measure spaces. Then there exists a unique measure $\mu_X\times\mu_Y$ on $\mathfrak{B}_X\times\mathfrak{B}_Y$ that ...
3
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0answers
35 views

Right-continuity of filtrations on product spaces

Let $(\Omega^1, \mathcal{F}^1)$ and $(\Omega^2,\mathcal{F}^2)$ be two measurable space and let $(\mathcal{F}^2_s)_{s \geq 0}$ be a filtration on $(\Omega^2,\mathcal{F}^2)$. Moreover, let $t\geq 0$ be ...
4
votes
3answers
142 views

Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$.

Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$. When I sketch these spaces it this statement makes sense intuitively because $[0, 1]\times[0, ...
5
votes
1answer
77 views

Is $(A \times B)^\epsilon \subseteq A^\epsilon \times B^\epsilon$?

While working on a problem related to my research, I had the following query. It pertains to product spaces: The Question: Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish (Complete separable metric) ...
1
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1answer
81 views

Are $\prod_{i\in I,j\in J}X_{ij}$ and $\prod_{i\in I}\prod_{j\in J}X_{ij}$ homeomorphic?

For each $i\in I, j\in J$, $X_{ij}$ is a topological space. Are $$\prod_{i\in I,j\in J}X_{ij}$$ and $$\prod_{i\in I}\prod_{j\in J}X_{ij}$$ homeomorphic? What's the homeomorphism?