# Questions tagged [simple-functions]

Use this tag for questions related to simple functions

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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} ...
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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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### 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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### 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 ...
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### 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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### 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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### 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 ...
### 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.