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Questions tagged [simple-functions]

Use this tag for questions related to simple functions

Simple functions are finite sums involving the characteristic functions of a set. They are real-valued functions on subsets of $\mathbb{R}$.

Simple functions are closed under addition and multiplication.

They are the foundation of Lebesgue integration (because the integration of simple functions is easy to define). A function $f$ can be approximated by simple functions allowing the calculation of the Lebesgue integral.

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How to prove that if $f$ is continuous a.e., then it is measurable.

Definition of simple function $f$ is said to be a simple function if $f$ can be written as $$f(\mathbf{x}) = \sum_{k=1}^{N} a_{_k} \chi_{_{E_k}}(\mathbf{x})$$ where $\{a_{_k}\}$ are distinct values for $k=1, 2, \cdots, N$…
Danny_Kim
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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.
FantasticAI
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Why do we require a function to be measurable in order to define its Lebesgue Integral?

Let be $f : X \to [0; + \infty)$ measurable. We define the Lebesgue integral for $f$ as follow: $$ \int_X f(x) \ d\lambda(x) := sup \{ \ \int_X s(x) \ d\lambda(x) : s(x) \le f(x) \ \forall x \in X \ \} $$ where s(x) is a non-negative simple…
Edoardo
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General set in product space approximated by rectangle sets

Let $(E^k,\mathcal{E}^k,\mu^k)$ be a product measure space. By a rectangle set in $E^k$, we mean a set of the form $A_1\times\ldots\times A_k$ where each $A_i\in \mathcal{E}$. My question is, for any $A\in \mathcal{E}^k$ such that…
Uchiha
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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
Strictly_increasing
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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}, \ \ \ \ x=\frac{a}{b} \text{ and } a + b =…
user808945
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simple function is dense in L^p space

I've been looking at the proof of 'simple function is dense in $L^p$ space in the Rudin's book 'Real and complex analysis(RCA)', Page 69, theorem 3.13. My question is in the assumption, there is a 'finite support simple function'. But the proof does…
kayak
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Approximation of a class of measurable functions by simple functions with "compact domain"

It is well known that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $f \ge 0$ then there exist a sequence of simple non-negative measurable functions {$ S_n $} such that $S_n\nearrow f$ but these simple functions have…
iki
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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 define for $f\in \Sigma$ that $$ \text{SF}(f)=\{s\in…
Ma Joad
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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 proof I've seen is the same. They all construct the…
Desura
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Tao's explanation on how to avoid disjointness for definition of simple function in derivation of Lebesgue Integral

Studying the Lebesgue Integral I am moving back and forth from different books (...I know, bad habit!), and I could not really figure out why sometime, when dealing with the definition of simple functions, some authors require the family of subsets…
Kolmin
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Why is the average of $\sin^2 wt = 1/2$ and $\cos^2 wt = 1/2$

Why is the average of both $\sin^2 = \frac{1}{2}$ and $\cos^2 = \frac{1}{2}$ I was revising Simple Harmonic motion notes and in the average of Kinetic energy derivation $$KE = \frac12 k A^2 \cos^2(\omega t)$$ And the solution is given as…
PsyScar
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Folland Theorem 6.14: Why is $ \lim |\int f_n g | \leq M_q(g)$?

My question comes from Theorem 6.14 of Folland's Real Analysis: 6.14 Theorem Let $p$ and $q$ be conjugate exponents. Suppose that $g$ is a measurable function on $X$ such that $fg \in L^1$ for all $f$ in the space $\sum$ of simple functions that…
Leonidas
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Lebesgue integration from definition

I'd like to ask you a question about finding integral value using the Lebesgue definition. I've been trying to find a method to obtain a sequence of simple functions that are convergent to function under integral on some integral in $ \\R $. It`s…
vearis
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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} where $F$ is some interval, $\gamma(\cdot | f)$ is a…
independentvariable
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