Questions tagged [simple-functions]

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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}{...
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1answer
24 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
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1answer
12 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 ...
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1answer
13 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}...
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1answer
33 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 \...
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1answer
22 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 ...
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0answers
15 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 ...
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1answer
48 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 ...
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0answers
23 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 ...
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1answer
83 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. $$ \...
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2answers
54 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 $x$ ...
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0answers
83 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 ...
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1answer
89 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 ...
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1answer
61 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 ...
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2answers
41 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 ...
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3answers
59 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 ...
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0answers
36 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 ...
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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.
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1answer
23 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:\...
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0answers
84 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{...
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1answer
387 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}...
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1answer
35 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 ...
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0answers
93 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
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1answer
36 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 ...
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6answers
78 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 $$f(x)=\sqrt{5-\frac{x^2}{x^2+2}}$$
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2answers
45 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 ...
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1answer
20 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=...
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1answer
327 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 ...
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2answers
58 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} \...
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1answer
572 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 ...
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1answer
40 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?
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1answer
115 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 ...
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0answers
91 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 =...
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1answer
247 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
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1answer
30 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 - \...
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1answer
286 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 ...
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1answer
68 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} \...
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1answer
59 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&...
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1answer
22 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 ...
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1answer
76 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 ...
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1answer
230 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?
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1answer
240 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
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1answer
534 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 ...
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0answers
31 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
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2answers
395 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 ...
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1answer
408 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: ...
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1answer
35 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\...
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1answer
2k 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.
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1answer
64 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 ...
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1answer
208 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 ...