Intereting Posts

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Centre of a matrix ring are diagonal matrices
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Why is $1, (x-5)^2, (x-5)^3$ a basis of $U=\{p \in \mathcal P_3(\mathbb R) \mid p'(5)=0\}$?
Finite abelian groups – direct sum of cyclic subgroup
Showing the countable direct product of $\mathbb{Z}$ is not projective
Product of two algebraic varieties is affine… are the two varieties affine?
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Fourier transform of a compactly supported function
If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} – x_{n_{k-1}}\| < 1/2^k$
Is every group a Galois group?
What is limit of $\sum \limits_{n=0}^{\infty}\frac{1}{(2n)!} $?
If a separately continuous function $f : ^2 \to \mathbb{R}$ vanishes on a dense set, must it vanish on the whole set?
Hamiltonian Path Detection

I have a homework problem that I’ve been stuck on for some time. Define $\mathcal{A}$ to be an algebra on $\mathbb{Q}$ generated by the set $S = \{ (a, b] \cap \mathbb{Q} : a, b \in \mathbb{Q} \}$, and define the function $\mu_0 : \mathcal{A} \to [0, \infty]$ by

$$ \mu_0 \left( \bigcup_{i=1}^{\infty} (a_i, b_i] \cap \mathbb{Q} \right) = \sum_{i=1}^{\infty} (b_i – a_i) $$

for a disjoint collection of sets $\{ (a_i, b_i] \cap \mathbb{Q} \} \subseteq S$. The question asks to show that $\mu_0$ is not a premeasure on $\mathcal{A}$.

First I see that

$$ \mu_0(\varnothing) = \mu_0 \left( \bigcup_{i=1}^{\infty} (i, i] \cap \mathbb{Q} \right) = \sum_{i=1}^{\infty} (i – i) = 0. $$

Also,

$$ \mu_0((a, b] \cap \mathbb{Q}) = b – a \quad (1) $$

by taking the union with infinitely many empty sets. So now for a disjoint collection of sets $\{ (a_i, b_i] \cap \mathbb{Q} \} \subseteq S$ whose union is in $\mathcal{A}$, why doesn’t $(1)$ imply we have $\sigma$-additivity:

$$ \mu_0 \left( \bigcup_{i=1}^{\infty} (a_i, b_i] \cap \mathbb{Q} \right) = \sum_{i=1}^{\infty} (b_i – a_i) = \sum_{i=1}^{\infty} \mu_0((a_i, b_i] \cap \mathbb{Q})? $$

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Let $(\varepsilon_n)_{n\in\mathbb{N}}$ be a sequence of rational numbers with $\sum_{n=1}^\infty\epsilon_n<1$, and let $(q_n)_{n\in\mathbb{N}}$ be an enumeration of $(0,1]\cap\mathbb{Q}$.

Define intervals $I_n$ for $n=1,2,\ldots$ as follows: If $q_n\in\bigcup_{k=1}^{n-1}I_n$, lelt $I_n=\emptyset$. Otherwise, let $I_n=(p_n,q_n]$ where $p_n$ is the largest of the numbers $0$, $q_n-\varepsilon_n$, $q_1$, … $q_{n-1}$ less than $q_n$.

Now $(0,1]\cap\mathbb{Q}$ is the disjoint union of the rational intervals $I_n\cap\mathbb{Q}$, and the sum of the lengths of these intervals is less than $1$.

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