# Questions tagged [borel-measures]

Use this tag for questions related to Borel measures, which, on a topological space, are measures defined on all open sets.

94 questions
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### How to decribe all bounded Borel functions where Lebesgue integral is zero

I'm working on an exercise and I'm not quite sure how to answer it. I need to describe all bounded borel measurable functions $f: [0,1] \to [-1,1]$ where $$\int_{[a,b)} f d\lambda = 0$$ There is a ...
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### Finding Borel sets

Consider a function $f:\mathbb R\to\overline{\mathbb R}$ defined as $$f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$$ Is $f$ Borel-measurable? I followed the answer given here ...
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### Lusin's theorem on arbitrary measure space

This question is generalization of Lusin's theorem, though it appears long. Let $(X, \rho)$ and $(Y, \sigma)$ be metric spaces with $(Y, \sigma)$ separable, and let $\mu$ be a finite Borel measure ...
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### Is the function $f(x) = \lfloor x\rfloor$ is measurable? [closed]

Let $(R, B)$, where $B$ is the Borel $\sigma$-algebra, be our measurable space. In this measure space how do you that the function $$f(x) = \lfloor x\rfloor$$ is measurable?
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### Tips to show that the Random Variable $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable

Let $X: \mathbb R \to \mathbb R$, where $\forall \omega \in \mathbb R -\mathbb Q: X(\omega)=0$. Show $X$ is $\mathcal{B}(\mathbb R)-\mathcal{B}(\mathbb R)-$measurable function: My ideas: Using the ...
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### Show $\mathcal{B}( \mathbb R)$ contains all countable subsets of $\mathbb R$

We defined $\mathcal{B}(\mathbb R):=\sigma(\{[a,b[\}),$ where $a < b \in \mathbb R.$ I have been able to prove: i) for any $b \in \mathbb R$, $\{b\} \in\mathcal{B}(\mathbb R)$ and subsequently ...
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### Lebesgue Measure equal to any translation invariant Measure on the Borel sigma Algebra

I have come across a question in my measure theory textbook and can pretty much figure out how to construct an answer to the second part but not the first part. Any solutions or hints would be ...
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### When sigma algebra of product topology equals sigma algebra of projections

Given a locally compact Hausdorff space $X$, let $S:= X^\mathbb{N}$ the corresponding product topology. I am trying to prove the following claim : $\sigma(S) = \sigma(p_i)$ where $\sigma(S)$ is the ...
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### A measure for which Lusin's theorem fails?

Lusin's theorem states that if $f:[a,b]\rightarrow\mathbb{R}$ is a Lebesgue measurable function, then for any $\epsilon>0$ there exists a compact subset $E$ of $[a,b]$ whose complement has Lebesgue ...
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### Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable?

Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable? Can one give a example of this ? I do really puzzled.
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### Finding a sequence of Borel measurable functions whose limit exist but Lebesgue integrals differ

I want to find a sequence of functions say $f_1 , f_2, \ldots$ such that each $f_n : [0,1] \to [0,+\infty)$ is a Borel measurable function with the limit $f(x)=\lim_{n \to \infty} f_n(x)$ existing for ...
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### Proof of factorization lemma

Let $X$ be a set, $(Y, \mathbb A)$ a measurable space and $g: X \to Y$ a surjective function. We consider on $X$ the $\sigma$-algebra induced by the function $g$ ($g^{-1}(\mathbb A))$. If the function ...
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### A continuous Function from $R$ to a Banach Space is Borel-Measurable

I think the notation of my book is a bit odd, so I'm having trouble finding any other sources to help me with this proof. The thing I'm trying to prove is that a continuous function, $f$, from $R$ to ...
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### Determining whether an interval (0,1] $\in\mathfrak B(\mathbb R)$ or not.

Determine, with justification, if the interval $(0,1]$ $\in\mathfrak B(\mathbb R)$ or not, given $\mathfrak B(\mathbb R)$ is the Borel $\sigma$-algebra on $\mathbb R$, which contains all the the open ...