# Questions tagged [simple-functions]

Use this tag for questions related to simple functions

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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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### 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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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 ...
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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### 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 ...
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 ...
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 ...
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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### 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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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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### 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 ...
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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### 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.
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 ...
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 ...