Questions tagged [simple-functions]

Use this tag for questions related to simple functions

Filter by
Sorted by
Tagged with
1
vote
0answers
28 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} ...
1
vote
1answer
79 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_{>...
1
vote
3answers
56 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 \...
0
votes
1answer
30 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 $\...
-1
votes
1answer
47 views

Proving a line is linear [closed]

If we have a simple function of a line $$f(x) = 2 + 3x$$ and want to prove it is linear, then wouldn't $$f(ax) = 2 + 3(ax) = af(x)$$? I am not seeing this to be the case
0
votes
1answer
32 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 ...
0
votes
1answer
39 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 ...
0
votes
0answers
23 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 $...
0
votes
1answer
96 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}:=\...
0
votes
0answers
61 views

Usamo 1980 Question

A two-pan balance is inaccurate since its balance arms are of different lengths and as pans are of different weights. Three objects of different weights A, B and C are each weighed separately. When ...
1
vote
1answer
53 views

The set of all functions that are uniform limit of simple functions

Theorem 4.19 of Bruckner's Real Analysis states that a bounded measurable function is uniform limit of simple functions; Wikiproof has a bit shorter proof. I was wondering if ($A=$) the set of all ...
3
votes
2answers
57 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(...
-1
votes
1answer
42 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 ...
0
votes
1answer
22 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?
0
votes
1answer
66 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 ...
1
vote
1answer
50 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 ...
0
votes
0answers
86 views

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, ...
0
votes
0answers
43 views

explicit form of function of simple functions

Define $X:\Omega\rightarrow \mathbb{R}$ to be simple if $X(\Omega)$ is a finite set. Then $X$ admits a representation $X(\omega)=\sum^n_{i=1} a_i 1_{A_i}$, where $a_i\in \mathbb{R}$, and $A_1, \dots, ...
0
votes
0answers
26 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 ...
-1
votes
1answer
29 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 ...
0
votes
1answer
85 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 ...
5
votes
1answer
386 views

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&...
-2
votes
1answer
57 views

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 ...
5
votes
1answer
92 views

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}...
0
votes
0answers
20 views

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}{...
-1
votes
1answer
36 views

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
0
votes
1answer
22 views

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 ...
0
votes
1answer
15 views

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}...
0
votes
1answer
77 views

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 \...
1
vote
1answer
23 views

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 ...
1
vote
0answers
22 views

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 ...
0
votes
1answer
58 views

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 ...
0
votes
0answers
45 views

Expanding Brackets Before Integrating a Linear Polynomial to the nth Power

I've just noticed that when integrating $$\int dx(x+a)^2 = \frac1 3(x+a)^3 + c = \frac1 3(x^3+3ax^2+3a^2x+a^3)+c$$ You have an $a^3 + c$ constant if integrated in the brackets, but if you expand ...
0
votes
1answer
255 views

approximation of 'any' bounded continuous function using bounded continuous functions with compact support

Suppose that $\phi$ is a bounded, continuous function with compact support $I$ (i.e. a bounded interval), then given any $\epsilon > 0$, there exists a simple function $\phi_{\epsilon}$ s.t. $$ \...
0
votes
2answers
173 views

Proof of surjection for piecewise $f: N \to N$

The problem gives the following piecewise function and asks for a proof of surjection. $f: N \to N$ as defined by $$f(x)= \begin{cases} x-1, & \text{if $x$ is odd,} \\ x+1, & \text{if $x$ ...
3
votes
0answers
250 views

Why don't we have "upper and lower" Lebesgue integrals?

For a function to be Riemann integrable, the upper and lower Riemann sum need to be equal. However, this no longer applies to Lebesgue integrals. Let $(\Omega,\Sigma,\mu)$ be a measure space and ...
1
vote
1answer
116 views

Epsilon/lambda of simple function

In my homework I have the following function: $u(x)=\sum_{n=1}^\infty \frac{1}{n^2(n+1)}1_{[-n,n]}$ I want to show that $\lambda(\{u\geq \epsilon\}) \leq \frac{2}{\epsilon}$ I have tried doing the ...
1
vote
1answer
63 views

Integral of complicated simple function

In my homework I am trying to find the integral of: $u(x)=\sum_{n=1}^\infty \frac{1}{n^2(n+1)}*1_[0,n]$ Using Excel, I can calculate that the integral is 1. However, when I try to show this ...
0
votes
2answers
43 views

Finding length of clothes

There are three types of clothes: A- Rs. 1 for 5mtrs B- Rs. 5 for 1mtr C- Rs. 1 for 1.5mtrs How much cloth is required for each type in mtrs, So that total for ...
0
votes
3answers
70 views

Show that $e^{-t^2} \in \mathcal{L}^1$

I have some homework where I need to determine if $u(t)=1/e^{t^2} \in \mathcal{L}^1$ on $(\mathbb{R},\mathcal{B}(\mathbb{R},\lambda))$ I have checked a theorem that says if it is Riemann integrable ...
1
vote
0answers
55 views

Approximating an integrable function with simple functions on compact sets

Let $f$ be an integrable function on $\mathbb{R}^n$. Then there exists a sequence of simple functions $\{f_n\}$ such that $\mid f_n \mid \leq \mid f \mid$ for all $n$ and $f_n$ converges to $f$ almost ...
1
vote
2answers
41 views

Find the value of $x +y$

If $a=\frac{x}{x^2+y^2}$ and $b=\frac{y}{x^2+y^2}$ then find $x+y$ I find that $x+y/y=\frac{a+b}{b}$ but the ans in the form of a and B only.
0
votes
1answer
46 views

When can we replace countably valued simple functions by finitely valued simple functions

Suppose that $(\Omega,\mathcal{A},\mu)$ is a finite measure space and $X$ is a Banach space. Let $f:\Omega \to X$ be a function that is an a.e. pointwise limit of countable-valued functions $f_n:\...
1
vote
0answers
88 views

Basic algebra for solving an equation, or not?

I have a "simple" equation which I have to solve for $ R_2 $: $$ \arccos\left(\frac{\sin\left(\arctan\left(\frac{M}{R_1+R_2-M}\right)\right)}{\sin\left(\frac{\pi}{T}\right)}\right)=\arctan\left(\frac{...
2
votes
1answer
1k views

Proving the Linearity of the Lebesgue Integral of Simple Functions

I don't understand a particular step in many proofs showing the linearity of the Lebesgue integral of simple functions. Consider the canonical decomposition of a simple function $\phi = \sum_{j=1}^{N}...
1
vote
1answer
51 views

Integral of continuous function over a triangle

Let $D\subset\mathbb{R^2}$ a triangle which has the corners $(0,0),(1,0),(0,1)$ and $g: \mathbb{R} -> \mathbb{R}$ continuous. Then $\int_Dg(x+y)dL^2(x,y)=\int_0^1tg(t)dt$ where $L^2$ is the ...
1
vote
0answers
163 views

Why is the indicator function of rational numbers a simple function?

I understand the simple function is one that is measurable and takes on finitely many values. However, the amount of values in the set Q are infinite. Thanks
-1
votes
1answer
43 views

Divide money and things equally between room mates [closed]

My friend and I spent 12730₹ in total to buy some house hold things(say some 10 items). We shared and paid equally. Now we are vacating so we have to divide the items, but my friend wants only 2 items ...
0
votes
5answers
82 views

How to find the domain and the range of this function$f(x)=\sqrt{5-\frac{x^2}{x^2+2}}$ algebraically? [closed]

help how to find the domain and the range of this function algebraically $$f(x)=\sqrt{5-\frac{x^2}{x^2+2}}$$
0
votes
2answers
49 views

Which mathematical law is used in $ab+ac-(b+c)=(a-1)(b+c)$

I just stumbled upon a question to figure out how to simplify J = (ab)+(ac)-(b+c) My steps: <=> a*(b+c)-b-c <=> a*(b+c) -1*(b+c) But that was not one of the solutions. One of these was, as ...