Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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879 views

An example of a generalized Cantor set with positive Lebesgue measure [duplicate]

I want to know if there exist a set $X\subset \mathbb R$ such that $X$ is $i)$ Perfect $ii)$ Compact $iii)$ Has empty interior $iv)$ Totally disconnected $v)$ Is not countable But $X$ has ...
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Basic question about the definition of an integral on a measure space

Let $(X,\mathcal{B},\mu)$ be a measure space. $\bf{\text{Definition:}}$ For a non-negative measurable function $f$ on $X$, $E\in \mathcal{B}$, $$\int_{E}f d\mu := \text{inf}\int_{E}\varphi d\mu$$ ...
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Riemann integral with intervals?

Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$ Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$. You can use the definition of Riemann integral ...
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281 views

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
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226 views

dominated convergence theorem

I am studying the proof of a theorem and in a part of the proof I have the following situation: Let $u : \Omega \rightarrow R$ a nonnegative measurable function, with $\Omega$ open and bounded. ...
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421 views

Lebesgue integral is linear in simple functions.

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $s,t: X\to[0,\infty)$ two simple functions. If $E\in\mathcal{M}$, show that, $$\int_E (s+t)\,d\mu=\int_E s\,d\mu+\int_E t\,d\mu$$ Attempt: By ...
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Proving that $\sigma(\tau_{\mathbb{R}}\times\sigma(\tau_{\mathbb{R}}))=\sigma(\tau_{\mathbb{R}})\otimes \sigma(\tau_{\mathbb{R}})$

$\sigma(\tau_{\mathbb{R}})$ denotes the Borel $\sigma$-algebra ($\tau_{\mathbb{R}}$ is the usual topology on $\mathbb{R}$), $\sigma(\tau_{\mathbb{R}}\times\sigma(\tau_{\mathbb{R}}))$ is the $\sigma$-...
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Open bounded set $E$ so that $m(E)\neq\lim_{n\rightarrow \infty}m(O_n)$

Let $E$ be a compact set and let us define the series: $$O_n=\{x\in R^d |d(x,E)<1/n\}$$ I proved that: $$m(E)=\lim_{n\rightarrow \infty}m(O_n)$$ Now I'm trying to find an open bounded set $E$ for ...
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How to recover a measure from its Fourier transform?

Let $f$ be the complex function defined on $\mathbb{R}$ by $$f(t)=\frac{1-it}{1+it}.$$ 1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
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1answer
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