# 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.

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### Complement of the support of measure on R^n has measure 0

How do we prove that $\mu (B) = 0$ for any ball $B$ such that $B \subset S^c$? For any $x \in S^c$, there exists $r_x > 0$ such that $\mu \left( B(x, r_x) \right) = 0$. Why there exists a countable ...
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### Whether or not a measure is finite or $\sigma$-finite

I have been given the following question: Let $\mu$ be a Borel measure on $[1, \infty)$ given by the density function 1/x with respect to the 1-dimensional Lebesgue measure. Is measure $\mu$ finite or ...
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### The space of Borel probability measures is closed in that of finite Borel measures w.r.t. Lévy–Prokhorov metric

I'm trying to prove this intuitive result. Could you have a check on my attempt? Let $(X, d)$ be a metric space. Let $\mathcal{M} :=\mathcal{M}(X)$ and $\mathcal{P} :=\mathcal{P}(X)$ be the sets all ...
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### Bijection $h:[0,1)\rightarrow [0,1)^{\mathbb{N}}$

I would like to find a referenceable source of the following cool technique to get an explicit bijection $h:[0,1)\rightarrow [0,1)^{\mathbb{N}}$ that is measurable and has measurable inverse: Cool ...
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### Show that the length of the union of intervals is less than (or equal to) the sum of the lengths of each of those intervals.

Let's define a generalised interval $\langle a,b \rangle$. $\langle a,b\rangle$ could be either one of the following: $[a,b], (a,b), [a,b), (a,b]$. Now let's define a length function on these ...
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### Show that $\mu \geq 0$ on Borel subsets of $X.$

Let $X$ be a compact Hausdorff space and $\mu$ be a complex Borel measure on $X$ such that $\int_{X} f\ d\mu \geq 0$ for every $f \in C(X)$ with $f \geq 0.$ Then show that $\mu$ is non-negative. ...
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### Is there a measurable function from $[0,1]$ to $ω_1$?

Does there exist a measurable function from $[0,1]$ (with the Lebesgue measure) to $ω_1$ that induces the Dieudonné measure? Definitions: $ω_1$ is the set of all countable ordinals, equipped with its ...
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### How is the measure of an arbitrary set calculated?

I am following Folland's Real Analysis text to learn measure theory and have so far been able to understand how Borel measures are constructed over $\mathbb{R}$ from increasing, right continuous ...
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