Questions tagged [measurable-functions]
For questions about measurable functions.
1,127
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Prove that a continuous and $\mathcal{F}_t$ adapted process is progressively measurable
I would like to show that if $X_t$ is a continuous and adapted stochastic process (real valued) then it is progressively measurable.
Here is my attempt : consider $B\in\mathcal{B}(\mathbb{R})$. Then ...
- 1,140
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72
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Bijection that takes continuum cardinal and null set to positive measure set
I've been dealing with the following problem:
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a bijection that verifies $m(f^{-1}(C)) > 0$ for $C$, a set with the continuum cardinal and $m(C)=0$, where $m$...
- 11
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How can I show that if $Y$ is $\mathcal{F}$ measurable then $X$ is also? [duplicate]
Let $X,Y:(\Omega, \mathcal{F}, \Bbb{P})\rightarrow (E,\mathcal{E})$ be random variables. Let us assume that $\mathcal{F}$ is a sigma algebra containing all $\Bbb{P}$-nullsets. I want to show that if $...
- 1,952
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1
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Uniform Convergence for Functions $X \to [0,+\infty]$?
An exercise in Tao's Introduction to Measure Theory asks to show that if $(X,\mathcal{B}, \mu)$ is a finite measure space and $f_n : X \to [0,+\infty]$ is a sequence of unsigned $\mathcal{B}$-...
- 2,129
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1
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The upper limit of a continuous function on two variables is measurable
Let $f(x,y):[0,1]\times[0,\infty)\to[0,1]$ be a continuous function. Let $$F(x)=\limsup_{y\to\infty}f(x,y).$$
Show that $F$ is a measurable function on $[0,1]$.
My attempts
In my limited experience, ...
- 65
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11
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Joint measurability of discrete integral.
I'm currently considering the following map: Let $\mathcal{F}$ be a subset of some $L^2$ space with respect to a probability measure on $\mathbb{R}$ and let $X=(X_1,X_2,\ldots,X_n)$ be points on the ...
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2
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Different definitions of the canonical form of simple functions (issue with constant reals different from $0$)
A simple function $\varphi$ is a finite sum of the form
$$\varphi (x) := \sum^{N}_{k = 1} r_k \mathbf{1}_{E_{k}} (x)$$
where $E_k$ is measurable for every $k \in \{1, \dots , N\}$ and $\{r_k\}^{N}_{k=...
- 3,991
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29
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Prove that $\{X_n\to X\}=\Omega\setminus \big\{\limsup_{n\to\infty}|X_n-X|>0\big\}$
Let $\Omega$ be measurable space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of real random variables. Denote by $\color{red}{\{X_n\to X\}}$ the set that contains exactly all elements $\omega\in \Omega$...
- 825
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$f_n \to f$ almost uniform if and only if $\lim_{n \to \infty} \mu ( \cup_{m \geq n} |f_m(x) - f_n(x)| \geq \epsilon ) = 0 $
Show that $f_n \to f$ almost uniform if and only if for all $\epsilon >0$ $\lim_{n \to \infty} \mu ( \cup_{m \geq n} |f_m(x) - f_n(x)| \geq \epsilon ) = 0 $
I have showed the first direction, but ...
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A Criterion that helps verify if a function $f:\Pi _{i=1}^nX_i\to Y$ is measurable
Let $\{X_i\}_{i=1}^n$ be a collection of measurable spaces and $Y$ a measurable space. Is there a criterion that helps verify if a function $f:\Pi _{i=1}^nX_i\to Y$ is measurable?
I know that a ...
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On the definition of locally integrable functions on an abstract measure space.
Recently I've been able to find a very cheap copy of the nice monograph [1], where I can find (chapter 10, §1, p. 163) the following general definition of Locally integrable function (respect to a ...
- 9,125
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Counterexample to Egorov for functions valued in non-separable metric space
A general form of Egorov (e.g. https://www.ime.usp.br/~glaucio/mat6704/textos/GMTLecureNotes.pdf) states that:
Egorov Theorem : Let $\mu$ be an outer measure on the set $X$ and $(Y,d)$ a separable ...
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Topology of convergence in measure is not compatible with the vector space structure of measurable functions
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
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Does Amann's Theorem 1.4 about $\mu$-measurability extend to metrizable topological groups?
Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space and $(E, | \cdot |)$ a Banach space. Let $f:X \to E$. We recall some definitions at page 62 of Amann's Analysis III.
$f$ is ...
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If $f \in L^0 (X, L^0 (Y))$, then $f \in L^0 (Z)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$\mu(X) < \infty$,
$(E, | \cdot |)$ a ...
- 5,128
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If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^0 (Y)$, then $(f_n)$ is Cauchy in $L^0 (Z)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
- 5,128
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0
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If $\| g_n \|_{L^0(X)} \to 0$ then $\| f_n \|_{L^0 (Z)} \to 0$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
- 5,128
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0
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21
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If $\nu(Y) < \infty$ then $F: X \to L^0(Y), x \mapsto f(x, \cdot)$ is $\mu$-measurable
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
- 5,128
0
votes
0
answers
20
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Convergence in a complete measure implies a.e. convergence for a subsequence
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions from $X$ ...
- 5,128
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How can i get that $\mathbb{E}[(\mathbb{E}(X|Y)-g(Y))(X-\mathbb{E}(X|Y))]=0$?
Let $X$, $Y$ be random variables in a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with $\mathbb{E}\left(X^{2}\right )<\infty$ and define the set:
$$\mathcal{N}:=\left\{g: g \text { is ...
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If $f \in L^0 (X, L^p_{\text{loc}} (Y))$, then $f \in L^0 (Z)$
Below we use Bochner measurability and Bochner integral. Let
$T>0$ and $p \in [1, \infty)$,
$X :=[0, T]$ and $Y:= \mathbb R^d$,
$\cal A$ the Lebesgue $\sigma$-algebra of $X$,
$\cal B$ the Lebesgue ...
- 5,128
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Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Then the map $x \mapsto \|f(x, \cdot)\|_{L^p_{\text{loc}}}$ is measurable
Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...
- 16.2k
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2
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Show that $T$ is not continuous when $p = \infty.$
Let $n \in \mathbb N,$ $1 \leq p \leq \infty$ and let $f \in L^p (\mathbb R^n).$ Define a function $T : \mathbb R^n \longrightarrow L^p (\mathbb R^n)$ by $$T (h) (x) = f(x + h)$$ for all $h \in \...
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If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$, then $(f_n)$ is Cauchy in $(L^0 (Z), \rho_Z)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
- 5,128
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0
answers
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The metric $\hat \rho (f, g) := \inf_{\delta >0} \{ \mu (|f - g| > \delta) +\delta \}$ on the space of $\mu$-measurable functions is complete
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
- 5,128
2
votes
0
answers
43
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Convergence in a complete measure is metrizable
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions from $X$ ...
- 5,128
1
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0
answers
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Measurable Riemann Mapping Theorem on a simply connected set
For context I am working through Sullivan's proof for the No-Wandering domain theorem.
My question is, can you restrict that Measurable Riemann Mapping Theorem to functions that are not defined on the ...
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Let $f \in L^0(X \times Y)$. Then for $\mu$-a.e. $x \in X$ we have $f(x, \cdot) \in L^0(Y)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
- 5,128
0
votes
0
answers
23
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A characterization of convergence in the metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
- 5,128
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votes
0
answers
29
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The metric space $(L^0 (X), \rho)$ of $\mu$-measurable functions is complete
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
- 5,128
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votes
0
answers
38
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Let $(f_n)$ be a Cauchy sequence in $S (X)$. There is a subsequence $(f_{n_k})$ and $f \in L^0(X)$ such that $f_{n_k} \to f$ a.e.
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions ...
- 5,128
0
votes
0
answers
20
views
Almost everywhere convergence in product measure and that in coordinate ones
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
- 5,128
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1
answer
42
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Are measurable functions surjective? [closed]
I learned that a measurable function can be defined as:
Let ( X, Σ ) and ( Y, T ) be two measurable spaces. A function f: X → Y is called measurable if for every E ∈ T the pre-image of E under f is ...
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For non second-countable TVS, is the sum of measurable functions again measurable? [duplicate]
Let $(\Omega, \mathcal A)$ be a measurable space and $E$ a topological vector space. Let $f,g:\Omega \to E$ be measurable. I already proved that
Theorem $E$ is second-countable, then $f+g$ is ...
- 16.2k
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Role of Compactness in Proof of Lusin's Theorem
In Tao's Introduction to Measure Theory, Lusin's theorem is presented and proved as follows: (I've adapted the proof slightly so it stands on its own.)
Theorem. Let $f : \mathbb{R}^d \to \mathbb{C}$ ...
- 2,129
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0
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Does a measurable function map every measurable set to a measurable set in its image? [closed]
So consider a measurable function $f: (X, \mathcal{F}) \rightarrow (Y, \mathcal{G})$.
The measurablility tells me that the pre-image of every measurable set $A\in \mathcal{G}$ is in $\mathcal{F}$, i.e....
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2
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$\int _0^1 f(x)g_n (x) dx \rightarrow 0 $ as $n\rightarrow 0$
Question:
Prove $\int _0^1 f(x)g_n (x) dx \rightarrow 0 $ as $n\rightarrow 0$ for $f\in \mathcal{L}^1([0,1])$ and $\{g_n\}_{n\in\mathbb{N}}$ a sequence of measurable functions on $[0,1]$ such that
(i) ...
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vote
1
answer
42
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Is the cardinality of $L^1[0,1]$ greater than $\frak c$?
I am looking for a normed space whose completion has strictly larger cardinality. I have settled on the space $I^1[0,1]$ of simple functions on $[0,1]$ with completion $L^1[0,1]$ the space of ...
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1
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68
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Does this condition about two measures $p$ and $q$ imply existence of $p$-integrable function that is not $q$-integrable?
Context :
I am trying to characterize some properties of barycenters of measures on a probability space, the following question arises. More precisely I want to restrict the support of a measure on ...
- 5,395
2
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1
answer
42
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measure theory: $g$ continuous a.e., $f_n \to f$ a.e., does $g \circ f_n \to g \circ f$ a.e.?
Question: Let $(f_n)_{n \geq 1}$, $f$, and $g$ be Borel measurable functions from $\mathbb{R} \to \mathbb{R}$. Suppose $g$ is continuous almost everywhere and $f_n \to f$ almost everywhere, does $g \...
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Proof of measurability in a proof
I am trying to understand a proof in a book (A. Weir, General Integration and Measure, p.111). There $\mu$-measurability of a function $f$ can be proved by checking if $\text{mid}(-g,f,g)$ is in $L^1$ ...
- 21
1
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1
answer
66
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Monotone convergence theorem on a function of two variables (integrating over one variable)
Consider a measurable space $\big( E \times F, \mathcal{E} \otimes \mathcal{F} \big)$, an $(\mathcal{E} \otimes \mathcal{F})$-measurable and positive function $f$, and a measure $\nu$ on $\big( F, \...
- 191
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0
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12
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Measurability of functions defined by integration of a two-variable function with respect to a non-$\Sigma$-finite transition kernel
Consider two measurable spaces: $\big( E, \mathcal{E} \big)$ and $\big( F, \mathcal{F} \big)$.
Consider a function $g \colon E \longrightarrow \mathbb{R_+}$ that is $\mathcal{E}$-measurable.
Consider ...
- 191
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22
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To prove convergence in L1 given that convergence in integral and absolutly convergence in integral
I am solving an exercise of real analysis:
Given that $f_n $ and $f$ is lebesgue integrable on $E$,$f_n\to f$ a.e. when $n\to\infty$.
If $$\lim_{n\to\infty}\int_Ef_n(x)\mathrm{d}x=\int_Ef(x)\mathrm{d}...
- 9
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25
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Whether the function $(x,x')\mapsto\rho\big(f(x),f(x')\big)$ is Borel for a Borel map $f\colon (X,d)\to (Y,\rho)$ with $(Y,\rho)$ not being separable?
Let $(X,d)$ be a compact metric space, let $(Y,\rho)$ be an arbitrary metric space. Let $\mathcal{B}(\mathbb{R}),\mathcal{B}(X),\mathcal{B}(Y),\mathcal{B}(X\times X), \mathcal{B}(Y\times Y)$ denote ...
- 411
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1
answer
24
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Continuous extension of compactly supported continuous functions
Suppose $D\subset \mathbb{R}^d$ is a path-connected compact (closed and bounded) domain. Let $C\subset D$ be a compact subset.
If the restriction $f|_C$ of a funciton $f: D\to \mathbb{R}$ is ...
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28
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$f: E \rightarrow \mathbb{R}$ is $\mathcal{E}$-measurable if and only if $f^{+}$ and $f^{-}$ are $\mathcal{E}$-measurable
Consider the measurable spaces $\left( E,\mathcal{E} \right)$ and $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$.
Prove that $f:E\rightarrow \mathbb{R}$ is $\mathcal{E}$-measurable $\iff$...
- 191
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0
answers
29
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Measurability with respect to complete filtration
I don't know if this is a difficult question, but I have absolutely no intuition about it.
Let us consider a probability space $(\Omega, \mathcal F, \mathbb P)$ supporting a $\mathbb R-$valued ...
- 31
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0
answers
64
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Density of Sequences of Hilbert Spaces and Bochner Spaces
Consider an integer $k\in\mathbb{N}$. Let $V_{k}, V, H$ be distinct Hilbert spaces with identical inner products and, therefore, the same norms. We have:
$$V_{k},V\subset H$$
In this context, I ...
- 320
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21
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Understanding part of a proof of dominated convergence Theorem from folland
The setting for DCT is some fixed measure space $(X,M,\mu)$
Proof. $f$ is measurable (perhaps after redefinition on a null set) by Propositions 2.11 and 2.12, and since $\left |f \right | \leq g$ a....
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