Questions tagged [simple-functions]
Use this tag for questions related to simple functions
135
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Need to use the standard representation of a simple function to prove Theorem 3.13? ("Measure, Integration & Real Analysis" by Sheldon Axler.)
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following theorem is Theorem 3.7 on p.76 in Section 3A in this book.
3.7 integral of a simple function
Suppose $...
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About Theorem 3.13 integral-type sums for simple functions in "Measure, Integration & Real Analysis" by Sheldon Axler.
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following theorem is Theorem 3.13 on p.79 in Section 3A in this book.
3.13 integral-type sums for simple ...
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Approximating the exponential function with simple functions
Let $f(x) = \mathrm{e}^{-x}\mathbb{1}_{[0, \infty)}$. I have a homework that asks me to approximate $f$ by a sequence of non-negative simple functions $f_n$ which is increasing and converges pointwise ...
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If $f:X\to \mathbb C$ is measurable simple function and $f\in L^p$, then $\mu(\{x\mid f(x)\neq 0\})<\infty$.
Does this proposition hold ?
I'm studying functional analysis and I come up with the situation where I (maybe) have to show this.
If $f:X\to \mathbb C$ is measurable simple function and $f\in L^p$, ...
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Easy Lebesgue integral of a non-horizontal line but by definition
Maybe a dumb question based on all the questions I've asked for the last decade, but what's the general way to do the Lebesgue integral of some non-negative (measurable?) function that is Riemann ...
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Show that a function is simple function
I was trying to calculate the Lebesgue integral of a function using $\int_S f\,d\mu = \sum_{k=1}^n a_k \,\mu(D_k\cap S)$, but I need to show that the function is indeed simple function first.
The ...
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How is $\xi+2a\eta<0$ an "obvious necessary condition" for $y^3+2y^2(1-2a-\xi)+y(1-4\xi+8a\xi)-2\xi-4a\eta >0$ to be satisfied for positive $y$?
How is $$\xi+2\alpha\eta<0$$ an 'obvious necessary condition' for the inequality
$$y^3+2y^2(1-2\alpha-\xi)+y(1-4\xi+8\alpha\xi)-2\xi-4\alpha\eta >0$$
to be satisfied for positive $y$ (as claimed ...
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Standard vs. regular representations of simple functions.
$\newcommand{\scrF}{\mathscr{F}}$
I'm trying to undertstand the difference between the standard vs. a regular representation for a simple function. The definitions I'm working with are as follows.
...
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Linear combination of simple function is a simple function
Let $f_i= \sum_{j=1}^m a_j\chi_{A_j}$ be a simple function. We have to show that finite linear cimbination of simple functions is a simple function.
$$ \sum_{i=1}^n \alpha_i f_i = \sum_{i=1}^n \...
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Density of L2 simple processes
Let us define the class of $\mathcal{L}^2\left(0,\infty\right)$ processes as the class of all processes $u$ that are progressively measurable and $\int_0^t\mathbb{E}\left[u_s^2\right]\,ds<\infty$ ...
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Why do we require a function to be measurable in order to define its Lebesgue Integral?
Let be $f : X \to [0; + \infty)$ measurable.
We define the Lebesgue integral for $f$ as follow: $$ \int_X f(x) \ d\lambda(x) := sup \{ \ \int_X s(x) \ d\lambda(x) : s(x) \le f(x) \ \forall x \in X \ \}...
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Verify whether this equation diagram is correct
I need to create an equation diagram representing a function written in python and I need help to verify that what I have created really means what I think it does. I am trying to say that if the ...
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Plot function with integral: find the right formula [closed]
How can I plot a function like this?
Where $N_\mathrm{A,0}$ is constant. For now, let's leave out the details: I just want to know what equation I should write as the argument of the ...
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How is $\dfrac{4x}{x-3}$ equal to $4+\dfrac {12}{x-3}$?
I am told that $\dfrac{4x}{x-3}$ is equal to $4+\dfrac {12}{x-3}$, but I have no idea how to arrive at that.
Can anyone, please, break it up for me?
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Question involving integration of nonnegative measurable functions using simple functions
Question: I am working on a problem in Folland's Real Analysis (specifically, 2.16) and I want to justify something:
Suppose $f$ is a measurable nonnegative function and $\int f<\infty$. Let $\...
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A question on the integral of nonnegative functions using the integral of simple functions
Question: In Folland's Real Analysis, on page $50$, he says, when discussing the integral of simple functions, that we can extend the integral of a simple function to all nonnegative functions $f$ by ...
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Show that $\phi_n(x) \leq \phi_{n+1}(x)$ (Folland Theorem 2.10)
I am reading the proof of Theorem 2.10 in Folland's Real Analysis.
I'm stuck to show the sentence:
It
is easily checked that $$\phi_n \leq \phi_{n+1}$$ for all $n$.
I first noticed that $E_{n}^{k}=...
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Question related to approximation of measurable functions by simple functions
I am self-studying measure theory from the book by Sheldon Axler.There I found a theorem before integration is introduced:
Let $(X,\mathcal S)$ be a measurable space and $f:X\to [-\infty,\infty]$ be a ...
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Folland Theorem 6.14: Why is $ \lim |\int f_n g | \leq M_q(g)$?
My question comes from Theorem 6.14 of Folland's Real Analysis:
6.14 Theorem Let $p$ and $q$ be conjugate exponents. Suppose that $g$ is a measurable function on $X$ such that $fg \in L^1$ for all $f$...
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Why proof of linearity of integration for simple functions requires non-negative coefficients?
In Folland's Real Analysis there's the following proposition in which (a) and (b) lead to the conclusion of linearity of integration for non-negative combinations:
Why do we require that $c\geq 0$? I ...
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Lebesgue integration from definition
I'd like to ask you a question about finding integral value using the Lebesgue definition. I've been trying to find a method to obtain a sequence of simple functions that are convergent to function ...
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Does there exist a constant $C>0$ such that for all simple functions $f$, $\int_1^\infty|f|\leq C(\int_1^\infty|f|^p)^{1/p}$?
I'm trying to solve the following problem.
Let $p\in(0,\infty)$ be fixed. Determine, with justification, whether the following statement is true or false.
There exists a constant $C>0$ such that ...
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Fubini's theorem for conditional measures
I have an integration that looks like:
\begin{align}\label{eq1}\tag{1}
\int_{f \in F} \left[\int_{x \in \mathbb{R}} \chi_{\{x \in A\}} \mathrm{d} \gamma(x|f)\right] \mathrm{d} \mu(f),
\end{align}
...
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Existence of Locally (Lebesgue-)Integrable Function
Given a locally integrable function $f: \mathbb R_{\geq0} \rightarrow \mathbb R_{\geq0}$, I wonder whether there exists an equivalent function that operates at a certain capacity $\nu\in\mathbb R_{>...
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Showing the absolute value of a simple function is a simple function:
Knowing that $f -g$ is a simple function I wanna show that $|f - g|$ is again a simple function.
Here is my trial: assuming that $ f = \sum_{i=1}^{n_1} a_i \chi_{A_i}$ and $g = \sum_{j=1}^{n_2} b_j \...
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A Question on Theorem 4.3 of Stein-Shakarchi [Approximating measurable functions with step functions]
I am confused on a part of the proof of Theorem 4.3 from Stein and Shakarchi's Real Analysis:
Theorem 4.3 Suppose $f$ is measurable on $\mathbb{R}^d$. Then there exists a sequence of step functions $\...
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L2 convergence for a simple function approximation
Consider the problem on the picture. I am struggling with part (b) of the excercise. I have managed to show that we have $L^1$ convergence, but I am unable to show $L^2$ convergence. Does anyone have ...
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Proof clarification measure given by integral
While studying measure theory, I encountered a simple proposition which contains a step I was not able to follow. First, here are some relevant definitions(not all of them, since there would be too ...
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Representation of Simple Function
Let $X$ be the universal set, and $E\subset X$ be a measurable set.
In Rudin, suppose the range of $s$ consists of the distinct numbers $c_1, ..., c_n$. Let $E_i =\{x|s(x)=c_i\} ~ (i=1,...,n)$. Then $...
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Simple functions dense in $C_{c}(X)$?
Let $X$ be a locally compact (Hausdorff) space and let $\mu$ be a (Radon) measure. Can compactly supported functions $X\to\mathbb{C}$ be approximated by simple functions w.r.t. the norm
$$\|f\|_{1}:=\...
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What if we take step functions instead of simple functions in the Lebesgue integral [duplicate]
When we define the Lebesgue integral, we first define it for simple functions $s(x) = \sum\limits_{j=1}^n c_j\chi_{A_j}(x)$ (where $A_j$ are measurable) as $\int sd\mu = \sum\limits_{i=j}^n c_j \mu(...
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Unique representation of simple function
Claim: Consider a simple function $f : X \to \mathbb{R}$ where $(X,M,\mu)$ is a measure space. If we represent $f$ as $\sum_{i=1}^{n} a_{i} \chi_{A_i}$, with $A_i \cap A_j = \varnothing$ and $a_i \ne ...
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Simplification of sum of exponentials? [closed]
Is there a way to simplify the expression $\sum _{k=0}^n 2^k$? That is, is there a way to write it without a $\sum$ or $\prod$ operator?
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Representation of the sum of simple functions
In the book Elementary Introduction to the Lebesgue Integral by Steven Krantz, the author considers two simple functions $\varphi$ and $\psi$ and says that the sum of these two functions has the ...
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Properties of Simple Functions
Suppose $f$ is a non-negative measurable function. i.e $f \in L^+$.
Is it true that f is the decreasing limit of a sequence of simple functions?
I'm suspecting that it's false, since it's we don't ...
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Horizontal truncation of the upper signed Lebesgue integral
Let $f:\mathbb R^d\to[0,+\infty]$. Prove or disprove that $$\lim_{n\to+\infty}\overline{\int_{\mathbb R^d}}\min(f(x),n)\,\mathrm{d}x=\overline{\int_{\mathbb R^d}}f(x)\,\mathrm{d}x.$$
I tried, ...
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Deterministic algorithm with math
Well, I’m not too good with math and I don’t know all of the special symbols, but I have one (maybe) simple goal to achieve:
How do I create a deterministic algorithm in math?
Well, the only input ...
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Is there 2 ways to write a parabolic equation, without facing your teacher's fury?
This is me being extremely stupid, but a parabola with a vertex $(2,6)$ and $x$-intercepts of $(-4,0),(8,0)$ can be described with the equation $y=\frac{-(x+4)(x-8)}{6}$(my answer) which would also be ...
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Riemann integrability of step functions
Suppose the step function is defined as follows. A function $f$ is a step function on $[a,b]$, if there exists a finite partition $P$ of $[a,b]$ such that $f$ is constant on the interior of each ...
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Riemann-Stieltjes integral of simple functions
I quote Øksendal (2003).
Let us consider a probability space
$\left(\Omega,\mathbb{P},\mathcal{A},\right)$ and a class of functions
$f:\left[0,\infty\right]\times\Omega\mapsto\mathbb{R}$. For $0\le
S&...
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If f and g are simple functions such that $f \le g$ then the Lebesgue integral $\int_Xfd\mu \le \int_Xgd\mu$
A common exercise I see in textbooks is the following:
If f and g are simple functions such that they are absolutely
convergent then: $$f \le g \Rightarrow \int_Xfd \mu \le \int_Xgd \mu$$
This video ...
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Is there a simple function $f(x)$ that follows $2$ rules when $x$ is rational?
Is there a simple function $f(x)$ that follows $2$ rules when $x$ is rational?
$x$'s simplest form is $\frac{a}{b}$ if $x$ is a rational number.
$$f(x) \in \begin{cases} \mathbb{R} \setminus \mathbb{Q}...
user808945
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the direction of >= sign when an equation is subjected to a condition
We have 2 equations
$$\frac{A}{I} = 2Tw$$
and
$$\frac{K+Z}{I} = w^2$$
and a condition that needs to be satisfied to find $A$:
where
$$T\geq 1$$
Does that mean $\frac{A}{...
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Multiplication of simple function looks like?
I was wondering what does the multiplication of two Lebesgue integrable simple functions look like. Assume integral of a function f is defined as
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How to deal with nonmeasurable sets when proving simple function is measurable?
I know the simple function f is defined as ai if x is in a sequence of measurable set Ei, 0 if it is not in the union of Ei.
I tried to prove that simple function is measurable.
It is measurable ...
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Confusion about the index of a sequence of simple functions, that approximates a measb. function.
It is known that a measurable function $f$ can always be approximated by a sequence of simple functions $f_n\uparrow f$ (pointwise). We can represent a simple function as
$g=\sum_{i=1}^mx_i\bf{1}_{A_j}...
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Show that $f$ is $\mathcal{A}$-measurable if and only if $\{x \in A \mid f(x)=\alpha_i\} \in \mathcal{A}$ for all $i=1,\dots,n$
Let $(X,\mathcal{A})$ be a measure space and $A \subseteq X$. Let $f:A \rightarrow [-\infty,+\infty]$ be a simple function, so that $\{f(x) \mid x \in A\}=\{\alpha_1,\dots,\alpha_n\}$ for some $n \in \...
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what does it mean by $\# \{ i : x_i \le t \}$?
I was reading through some manuscript and wanted to try to implement some stuff mentioned in the text. I came across this equation
equation $= \# \{ i : x(i) \le t \}$
I went through a couple of ...
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Integral when limit of nondecreasing simple functions greater than another simple function
This looks intuitive but I have a problem writing the proof.
Let $(f_n)$ (nondecreasing) and $f$ all be nonnegative simple functions in a measure space. If $\lim f_n\geq f$ pointwise, then $\lim\int ...
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Integral of simple functions and the convention $0 \times \infty = 0$
I am studying measure theory on my own and there is something about the convention that $0 \times \infty = 0$ that I can's seem to get my head around. I've read various threads now on this topic and ...
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