# Questions tagged [simple-functions]

Use this tag for questions related to simple functions

77 questions
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.
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### 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 ...
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 ...
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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?
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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.
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### 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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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 ...
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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: \...
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 ...
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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 ...
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 $A_i$...
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 $$9x^2+8y^2-36x-16y-28=0$$
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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 ...
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### 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 ...
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 ...