Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [simple-functions]

Use this tag for questions related to simple functions

1
vote
2answers
38 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
12 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
81 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{...
1
vote
1answer
47 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
26 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 ...
1
vote
0answers
48 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
29 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
6answers
72 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
41 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 ...
1
vote
1answer
14 views

Prove simple closed curves $f$'s exist, so $\Gamma = C-\sum_{i=1}^{k}{f_i}$ satisfies $ \int_{\Gamma}{\frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$

Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $k\in \mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, \dots ,f_k$ such that the cycle $\Gamma = C-\sum_{i=...
0
votes
1answer
65 views

A simple function and its canonical form.

Simple functions are of the form $\phi(x) = \sum_{k=1}^N a_k \chi_{A_k}(x)$ where $\chi$ is the indicator function and that $A_k$'s are measurable sets. This is how Stein defines a simple function ...
1
vote
2answers
34 views

Equality of simple functions with measures.

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $a_i, b_j \geq 0$ and $A_i, B_j \in \mathcal{M}$ of finite measure, for $1 \leq i\leq n$, $1 \leq j \leq m$. If $\sum_{i=1}^n a_i\chi_{A_i} \...
0
votes
1answer
130 views

Showing that Continuous functions are dense in L1 by use of $\sigma$-algebra.

I have a rather vague question, but, my analysis teacher left as an exercise to show that the continuous functions are dense in L1. There are other way to do this, but she specified that this must be ...
0
votes
1answer
34 views

How to find $n$ in this equation? (involving modulus)

How to find $n$ in this equation? $10 \le (7n) \mod 24 \le 13$ Can I use program to solve this kind of equation?
3
votes
1answer
42 views

Continuity implies $\mu$-strongly measurability?

In view of the definitions below (That can be found in Infinite Dimensional Analysis: A Hitchhiker's Guide, of ALIPRANTIS and BORDER): Definition 1. Suppose $\Omega$ is a set equipped with an álgebra ...
0
votes
0answers
31 views

Clarifying Textbook's Example of Integral of a Simple Function

Definition 88.1 Let $s$ be a nonnegative simple measurable function in $s =\sum_{i=1}^n c_i \chi_{A_i}$, where $A_i \in \mathscr{M}$ and $c_i \geq 0$ for $i = 1, \dotsc, n$. We define the ...
0
votes
0answers
40 views

How to build a simple Mathematical formula with matching condition

I'm trying to write a mathematical equation that sums a set of variables on the condition that they match variables in an original array. Example: List A: A1 = Dog, A2 = Cat, A3 = Monkey List B: B1 =...
1
vote
1answer
66 views

Lebesgue Integrability of $x^{-p}$

This answer (https://math.stackexchange.com/a/1540107/273275) uses $$f(x) = x^{-p},\qquad x\geq1$$ is lebesgue integrable for $p>1$. But I did not manage to prove it. $f$ is obviously measurable. ...
2
votes
1answer
29 views

Approximating measureable functions by simple functions on measureable sets.

My Professor stated the following claim without proof: If $f$ is measureable, then for all $\epsilon > 0$, there exists $E$ and a simple function $\phi$ such that $\mu(E) < \epsilon$ and $|f - \...
-1
votes
1answer
147 views

Uniqueness in the decomposition of a simple function in the canonical form.

I know that every simple function $\varphi$ is a finite sum $$\varphi (x) = \sum_{k=1}^N a_k \chi_{E_k} (x).$$ where the $E_k$ are measurable sets of finite measure and $a_k$ are constants.This ...
0
votes
1answer
49 views

Simple question on simple functions: interpreting the sum sign

Definition: If $f$ is a simple function with range $R(f)=\{a_1,a_2,\dots,a_n\}$ and $E_k=\{x\in D(f):f(x)=a_k\}$ for $k\in \{1,2,...,n\}$ then the Canonical Representation of $f$ is $$\begin{align} \...
0
votes
1answer
47 views

show that non degeneracy sets are equipotent between them.

I have to prove it for these sets and find a bijection: a) $(a,b) \sim [a,b] \sim [a,b) \sim (a,b]$ where: $a,b\in R$ with $a<b$ b) $[a,b] \sim [c,d]$ where: $a,b,c,d \in R$ with $a<b ,c&...
0
votes
1answer
21 views

Calculate how many of a compound discounting/increasing asset i can purchase with a set amount (with limited equation

We're running a financial calculation in a simple programming language with limited built-in math functions. We need to build two equations, one for purchasing an asset that increases slightly in ...
1
vote
1answer
51 views

How to prove convergence of a sequence of binary numbers

I have a boolean expression with 4 inputs and 1 output, that when iterated onto itself(output->input(s)), the function converges to 1. How do I go about proving the convergence of a sequence of binary ...
1
vote
1answer
110 views

Prove that a simple connected graph has even numbers of vertex [closed]

Given a simple connected graph G that all of its vertex degree is 3. how can I prove that G has even number of vertex? and does G has a perfect matching and why?
-3
votes
1answer
173 views

Prove that $1+{1\over 1+{1\over 1+{1\over 1+{1\over 1+…}}}}=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+…}}}}$ [duplicate]

So, my professor me gave this exercise as a challenge: -First, prove that: $$1+{1\over 1+{1\over 1+{1\over 1+{1\over 1+...}}}}={1+\sqrt{5}\over 2}.$$ -Then, prove that: $$1+{1\over 1+{1\over 1+{1\...
2
votes
1answer
357 views

If $f$ is integrable, $g$ measurable, and $f = g$ a.e., then $g$ is integrable

The precise statement is if $f$ is integrable, $g$ measurable, and $f = g$ almost everywhere, then $g$ is integrable, and the integrals coincide. I use the following definition of integrable: $f$ is ...
0
votes
0answers
29 views

Multiplying out squared brackets

This seems like a very simple problem, but I am getting confused. When you multiply out say $(x+4)^2$ then it is the same as $x(x + 4) + 4(x + 4)$, so I believe the correct output would be $x^2 + 8x +...
2
votes
1answer
151 views

Lebesgue integration of step function

Let $J$ be a finite subinterval of the real line and $f:J\rightarrow\mathbb{R}$ a simple function taking on values $c_1,...,c_n$. The function $f$ is called a step function if $f^{-1}(c_i)$ is a ...
0
votes
1answer
205 views

Understanding $a_i$'s in simple functions in measure theory

In search for help here. Will appreciate any support from you all. Thanks in advance. So, I have already checked a couple of questions I found here: Understanding Simple Functions and: ...
1
vote
1answer
31 views

About the converge of simple functions

I am now considering such a problem. Suppose $f\in L^1([0,1])$, for any integer $n$, we can construct such a measurable function: $$Pn(f)=n\sum\limits_{k=1}^n\int_\frac{k-1}{n}^\frac{k}{n} f d\lambda\...
2
votes
1answer
668 views

How to prove simple function is measurable

I'm aware of the definition of the measurable function. But I was wondering how to prove simple function is measurable? It would be better have some detailed proof.
-2
votes
1answer
57 views

Union of two elements in a sigma algebra

Let $f$ and $g$ are two non-negetive simple functions on $X$. Then show that the set $A$ belongs to $£$, where $A=\{x:f(x)>=g(x)\}$ and $£$ is the sigma algebra of subsets of $X$. Also I stuck ...
-1
votes
1answer
125 views

Expectation of a random variable - Proof using MCT and simple functions [closed]

I got the following task: Let $(\Omega,F,P)$ be a probability space and $X:\Omega\rightarrow R$ a random variable with $X\geq0$ P-a.s. Prove that $E[X]=0$ implies that $X=0$ P-a.s. I tried to prove ...
2
votes
1answer
282 views

Definition of a Stochastic Integral for Simple/Elementary Stochastic Processes being well defined

Often authors in stochastic calculus books define the set of all "simple" or elementary stochastic processes to be the set of all functions $H:\Omega\times[0,1]\longrightarrow \mathbb{R}$ such that: \...
2
votes
0answers
134 views

Approximate a simple function with the sum of two simple functions

Let $(\Omega, \mathscr F, \nu)$ be a measure space and let $f$ and $g$ be non-negative Borel functions. Let $\psi$ be a simple function such that $0 \leq \psi \leq f + g$. I want to show that it is ...
4
votes
1answer
133 views

Non constructive proof that positive measurable functions can be approximated by a sequence of simple functions?

Let $f \geq 0$ be measurable. Then $\exists \ 0 \leq \phi_n \leq f$ an increasing sequence of simple, measurable functions such that $\phi_n \rightarrow f$ as $n$ goes to $+\infty$. Every single ...
3
votes
1answer
119 views

Is every monotonic simple function a step function?

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a simple function, that is, $f (x) = \sum_{i = 1}^{n} a_i \mathbb{1}_{A_i}(x)$ for all $x \in \mathbb{R}$, where each $a_i$ is a constant, each $A_i$...
-1
votes
1answer
472 views

Find the minimum and maximum distances from point $(2,6)$ to ellipse $9x^2+8y^2-36x-16y-28=0$

Having trouble with this. I'm not getting any ideas. Find the minimum and maximum distances of point $(2,6)$ from the ellipse $$9x^2+8y^2-36x-16y-28=0$$
1
vote
2answers
445 views

Approximating a continuous function with compact support by simple functions from above?

Let $f\in C_c(X), f:X \to \mathbb{R}$ be a continuous function with compact support on a measure space $X$ with $f \ge 0$. Allegedly it is possiple to approximate that function by a monotonic ...
0
votes
1answer
28 views

Is it true that $\exists a\in I\ \text{and}\ \epsilon>0$ such that $f$ is constant on $(a-\epsilon,a+\epsilon)\bigcap I$?

Let $f:I\to\mathbb{R}$ be a simple function where $I$ is an interval with more than one point. Is it true that $\exists a\in I\ \text{and}\ \epsilon>0$ such that $f$ is constant on $(a-\epsilon,a+\...
2
votes
0answers
51 views

Let $\{a_n\}_{n=1}^{\infty}\subset [0,1]$ be equidistributed and set $\mu_N=\tfrac 1N\sum_{n=1}^N\delta_{a_n}$. Then $\int_0^1fd\mu_N\to\int_0^1fdm$

I'm working on a question, stated as follows: We say that a sequence $\{a_n\}_{n=1}^{\infty}$ in $[0,1]$ is equidistributed (in $[0,1]$) if and only if for all intervals $[c,d]\subset [0,1]$, $$ \...
3
votes
1answer
332 views

Existence of step function approximation for Lebesgue function in norm by Littlewood's principle

The original question: For any $f \in L_1[a,b]$, there exists a sequence of step function $h_n$ such that $\lim \int^a_b |h_n-f| = 0$ My approach is using littlewood principle, for any $\epsilon &...
0
votes
0answers
172 views

Any step function on $[a,b]$ is a simple function.

My work: To show this, I consider the function $\varphi:E\rightarrow R$ so that $\varphi(E)=\{c_{1},..c_{p}\}$ is a finite set and if $c_{1},..c_{p}$ distnict, then $\varphi = \sum_{j=1}^{p}c_{j}\chi_{...
0
votes
0answers
29 views

What is the exact meaning of mean, and what is the easy way of getting it? I've heard two versions.

I am studying for Pre-GCSE maths paper and I need to learn Mode, Mean, Medium, Range and PTask. I have learnt all of them except the mode, infact I have learnt mode but I have heard two versions of it....
3
votes
2answers
56 views

Maths Question, not sure how to explain it.

I am a student that is going to be taking his Pre-GCSE in School soon, I am not very good with Maths but I have been watching a few videos online trying to self teach myself. I have been good so far ...
3
votes
1answer
223 views

Definition of Lebesgue Integral, why define integral for bounded functions? [Stein]

Stein's development for integral: (1.) Define the integral of simple measurable functions. (1'). Define the integral of bounded measurable functions on sets of finite measure with (1.) (2.) ...
1
vote
1answer
131 views

an almost everywhere limit of simple functions is a uniform limit of countably valued functions.

$(\Omega,\Sigma,\mu)$ is a probability measure space, and $X$ is a Banach space. $f_n:\Omega\rightarrow X$ are simple functions for each $n\in\mathbb{N}$, and$f:\Omega\rightarrow X$ satisfies $f_n\...
0
votes
2answers
103 views

Function to change input from 0 - 100 to 50 - 100

I'm looking to write a function so when I input 100 I get 50 out on the other end. Although, when I input 0, I'd like to get out 100 on the other end. Besides this, I need to be able to input ...
1
vote
1answer
43 views

$||f||_p <\infty$ , then there exist a sequence of simple functions $\{g_n\}$ such that $||g_n-f||_p\to 0$?

Let $f$ be a measurable function on a measure space $(X,\mathcal F,\mu)$ and $p>0$ such that $||f||_p:=(\int_X|f|^p d\mu)^{1/p} <\infty$ , then is it true that there exist a sequence of simple ...