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Questions tagged [simple-functions]

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Approximation of a class of measurable functions by simple functions with "compact domain"

It is well known that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $f \ge 0$ then there exist a sequence of simple non-negative measurable functions {$ S_n $} such that $S_n\...
iki's user avatar
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Why have simple function only a finite number of summands?

I have a short question about simple functions which are used to define the Lebesgue-Integral. Is there a special reason, why we only allow a finite number of summands in defining simple functions? ...
RobRTex's user avatar
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Construction of a simple function by open sets

A simple function $\phi: \mathbb{R} \to \mathbb{R}$ can be written as: $$\phi(x) = \sum_{k=1}^n a_k \chi_{E_k}(x)$$ where each $E_k$ is a measurable set and $\cup E_k = \mathbb{R}$ and $\chi_{E_k}$ is ...
MC2's user avatar
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Detail in standard measure theory I cannot seem to obtain

There is a standard result in measure/integration theory which I just cannot seem to obtain. If $f \colon X \to \mathbb{C}$ is measurable ($X$ is any measurable space), there exist simple measurable ...
Qeeko's user avatar
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20 views

Hardy-Littlewood maximal function algorithm

There exists implemented algorithm which compute the Hardy-Littlewood maximal function at least for reasonably simple non-trivial cases? E.g. piecewise linear functions?
Giafazio's user avatar
  • 330
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0 answers
19 views

Simple functions convergence under different topologies

In Serge Lang's Real and Functional Analysis, first part of Lemma 3.1 (p.129) states Let $\{f_n\}$ be a Cauchy sequence of step mappings. Then there exists a subsequence which converges pointwise ...
user760's user avatar
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2 votes
2 answers
149 views

Divergence for original function implies large value eventually for some simple function approximation

Let $g\in L^1[0,1]$ with $\|g\|_{L^1[0,1]}\leq1$. We are given a sequence of continuous, linear functionals $f_n$ such that $f_n(g)\to\infty$ as $n\to\infty$. Now given some $L>0$, I'm wondering ...
Václav Mordvinov's user avatar
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1 answer
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Can we compare simple functions without looking at their values at specific points?

Recently I realized that measurable functions can always be written as countable sum of indicator function of measurable sets. This is because we can write for $n\in\mathbb Z$, $A_n=\bigcup_{k=-\infty}...
P. Quinton's user avatar
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68 views

Show $f: X \rightarrow \mathbb R^n$ is measurable iff condition holds

Let $(X, \mathcal E, \mu)$ be a $\sigma$-finite measure space. Prove: A function $f: X \rightarrow \mathbb R^n$ is measurable if and only if there exists a sequence of simple, measurable functions $...
Minerva's user avatar
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Uniform convergence of increasing sequence of simple functions

Let $\mu$ be a measure on $\mathbb{R}^n, \Omega \subseteq \mathbb{R}^n$ be $\mu$-measurable. Let $f:\Omega \rightarrow \mathbb{R}^n$ be $\mu$-measurable, $\forall j \in \mathbb{N}: A_j \subseteq \...
strugglingStudent's user avatar
3 votes
1 answer
433 views

Why is the average of $\sin^2 wt = 1/2$ and $\cos^2 wt = 1/2$ [closed]

Why is the average of both $\sin^2 = \frac{1}{2}$ and $\cos^2 = \frac{1}{2}$ I was revising Simple Harmonic motion notes and in the average of Kinetic energy derivation $$KE = \frac12 k A^2 \cos^2(\...
PsyScar's user avatar
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-2 votes
3 answers
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Is there a simple function mapping -1 to 1 and 1 to 0 [closed]

This is a rather simple question. I'm writing a computer program, and I have a variable for velocity, which takes values $\pm 1$. I also have corresponding sprites, facing both right and left, on ...
Math chiller's user avatar
2 votes
2 answers
181 views

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 $...
tchappy ha's user avatar
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1 vote
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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 ...
tchappy ha's user avatar
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1 vote
2 answers
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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 ...
ADotByMyName.'s user avatar
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1 answer
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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 $f\in L^p$, ...
daㅤ's user avatar
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1 answer
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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 ...
BCLC's user avatar
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-1 votes
1 answer
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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 ...
dddu1qa's user avatar
-3 votes
1 answer
48 views

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 ...
Tlotlo Oepeng's user avatar
1 vote
0 answers
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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. ...
Irving Rabin's user avatar
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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 \...
nodis6's user avatar
  • 121
5 votes
4 answers
587 views

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 \ \}...
Edoardo's user avatar
  • 53
0 votes
1 answer
29 views

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 ...
berimbolo's user avatar
  • 111
1 vote
1 answer
46 views

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 ...
user3713179's user avatar
-2 votes
1 answer
51 views

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?
brilliant's user avatar
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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 $\...
User7238's user avatar
  • 2,534
0 votes
1 answer
37 views

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 ...
User7238's user avatar
  • 2,534
2 votes
1 answer
232 views

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}=...
Math's user avatar
  • 2,379
0 votes
1 answer
119 views

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 ...
Kishalay Sarkar's user avatar
2 votes
1 answer
172 views

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 $f$...
Leonidas's user avatar
  • 1,054
0 votes
1 answer
228 views

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 ...
Anon's user avatar
  • 1,791
3 votes
1 answer
92 views

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 ...
vearis's user avatar
  • 53
0 votes
2 answers
85 views

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 ...
omololo's user avatar
  • 285
3 votes
0 answers
287 views

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} ...
independentvariable's user avatar
1 vote
1 answer
154 views

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_{>...
Michael's user avatar
  • 365
1 vote
3 answers
352 views

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 \...
user avatar
1 vote
1 answer
397 views

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 $\...
Leonidas's user avatar
  • 1,054
0 votes
1 answer
368 views

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 ...
PROB123's user avatar
  • 103
0 votes
1 answer
62 views

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 ...
Lucas's user avatar
  • 646
0 votes
0 answers
212 views

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 $...
maskeran's user avatar
  • 573
0 votes
1 answer
339 views

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}:=\...
Calculix's user avatar
  • 3,386
3 votes
2 answers
149 views

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(...
edamondo's user avatar
  • 1,397
0 votes
1 answer
164 views

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 ...
sixtyTonneAngel's user avatar
0 votes
1 answer
25 views

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?
user56834's user avatar
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0 votes
1 answer
464 views

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 ...
Math's user avatar
  • 2,379
2 votes
1 answer
153 views

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 ...
Toasted_Brain's user avatar
0 votes
1 answer
182 views

Horizontal truncation of the upper signed Lebesgue integral

Prove or disprove that $\forall d\ge_\mathbb Z0,\forall f\colon\mathbb R^d\to[0,+\infty],$ $$\lim_{n\to+\infty}\overline{\int_{\mathbb R^d}}\min(f(x),n)\,\mathrm{d}x=\overline{\int_{\mathbb R^d}}f(x)\,...
Abraham Zhang's user avatar
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0 answers
34 views

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 ...
Carter's user avatar
  • 1
-1 votes
1 answer
34 views

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 ...
uSaUCyBOI's user avatar
0 votes
1 answer
214 views

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 ...
Guangyao's user avatar
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