# Blueprint for the Liquid Tensor Experiment

## 2.2 Spaces of Measures

Definition 2.2.1
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Let $$0 {\lt} p' {\lt} 1$$ be a real number, and let $$S$$ be a finite set. Then $$\mathcal M_{p'}(S)$$ denotes the real vector space

$\mathcal M_{p'}(S) = \left\{ \sum _{s\in S}a_{s}[s] \text{ such that } a_{s}\in \mathbb {R}\right\}$

endowed with the $$\ell ^{p'}$$-norm $$\Vert \sum _{s\in S}a_s[s]\Vert _{\ell ^{p'}}=\sum _{s\in S}\lvert a_s\rvert ^{p'}$$.

For every finite set $$S$$, the space $$\mathcal M_{p'}(S)$$ can be written as the colimit (or simply the union, in this case)

$\mathcal M_{p'}(S) = \varinjlim _{c{\gt}0}\mathcal M_{p'}(S)_{\leq c}$

where $$\mathcal M_{p'}(S)_{\leq c} = \left\{ F \in \mathcal M_{p'}(S) \text{ such that } \Vert F \Vert _{\ell ^{p'}}\leq c\right\}$$. Now, given a profinite set $$S=\varprojlim S_i$$, where all $$S_i$$’s are finite sets, and a positive real number $$c{\gt}0$$, define $$\mathcal M_{p'}(S)_{\leq c}$$ as

$\mathcal M_{p'}(S)_{\leq c}=\varprojlim \mathcal M_{p'}(S_i)_{\leq c}$

endowed with the projective limit topology. Finally, set $$\mathcal M_{p'}(S)=\varinjlim _{c}\mathcal M_{p'}(S)_{\leq c}$$.

Definition 2.2.2
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Let $$0 {\lt} r' {\lt} 1$$ be a real number, and let $$S$$ be a finite set. Then $$\mathcal L_{r'}(S)$$ denotes the group

$\mathcal L_{r'}(S)=\{ \sum _{s\in S,n\in \mathbb {Z}}a_{n,s}T^n[s] \mid a_{n,s}\in \mathbb {Z}, \sum _{n\in \mathbb {Z}, s\in S}\lvert a_{n,s}\rvert r^n {\lt} + \infty \} .$

The group $$\mathcal L_{r'}(S)$$ is filtered by the subsets

$\mathcal L_{r'}(S)_{\leq c}=\bigl\{ \sum _{s\in S,n\in \mathbb {Z}}a_{n,s}T^n[s] \mid a_{n,s}\in \mathbb {Z}, \sum _{n\in \mathbb {Z}, s\in S}\lvert a_{n,s}\rvert r^n \leq c\bigr\}$

for $$c{\gt} 0$$. Each subset $$\mathcal L_{r'}(S)_{\leq c}$$ can be written as

$\mathcal L_{r'}(S)_{\leq c}=\varprojlim _{A}\mathcal L_{r'}(S)_{A,\leq c}$

where $$A$$ runs through the finite subsets of $$\mathbb {Z}$$ and $$\mathcal L_{r'}(S)_{A,\leq c}$$ is the finite set

$\mathcal L_{r'}(S)_{A,\leq c}=\bigl\{ \sum _{s\in S,n\in A}a_{n,s}T^n[s] \mid a_{n,s}\in \mathbb {Z}, \sum _{n\in A, s\in S}\lvert a_{n,s}\rvert r^n\leq c\bigr\} .$

This defines a profinite topology on $$\mathcal L_{r'}(S)_{\leq c}$$, for each $$c{\gt}0$$.

Definition 2.2.3
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Let $$0 {\lt} r' {\lt} 1$$ be a real number, and let $$S$$ be a profinite set with a presentation $$S=\varprojlim S_i$$, where all $$S_i$$ are finite. For all $$c{\gt}0$$, let $$\mathcal L_{r'}(S)_{\leq c}$$ denote the projective limit

$\mathcal L_{r'}(S)_{\leq c}=\varprojlim \mathcal L_{r'}(S_i)_{\leq c},$

endowed with the projective limit topology, and set

$\mathcal L_{r'}(S)=\varinjlim _{c {\gt} 0}\mathcal L_{r'}(S)_{\leq c}.$

Now fix $$0{\lt} \xi {\lt}1$$ and let $$x\in \mathbb {R}_{\geq 0}$$. For simplicity, denote by $$\mathbb {Z}(\! (T)\! )_{r'}$$ the group $$\mathcal L_{r'}(\ast )$$. One of the key results proven in [ Sch20 , § 6 ] is the surjectivity and continuity of the following map.

Definition 2.2.4
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Fix $$0{\lt}\xi {\lt}1$$, and let $$\vartheta _{\xi }\colon \mathbb {Z}(\! (T)\! )_r\rightarrow \mathbb {R}$$ be the evaluation map

$\sum a_nT^n\longmapsto \sum a_n\xi ^n.$

In order to prove the surjectivity, we use the following construction.

Definition 2.2.5
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For all $$x \in \mathbb {R}$$ and $$n\in \mathbb {N}$$, set

$Y_\xi (x, n)= \begin{cases} x& \text{ if } n = 0 \\ Y_\xi (x,n-1) - \left\lfloor \frac{Y_\xi (x,n-1)}{\xi ^{n-1}} \right\rfloor \xi ^{n-1} & \text{ if } n \ge 1 \end{cases}$

where $$\lfloor \ast \rfloor$$ denotes the floor of $$\ast$$, namely the greatest integer that is less or equal than $$\ast$$.

Lemma 2.2.6
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The map $$\vartheta _{\xi }$$ is surjective.

Proof

Pick $$x\in \mathbb {R}$$, and consider the power series

$F(T)=\sum _{n\geq 0}\left\lfloor \frac{Y_\xi (x,n)}{\xi ^n}\right\rfloor T^n\in \mathbb {Z}[\! [T]\! ]$

where $$\{ Y_\xi (x,n)\} _{n}$$ is the sequence defined in Definition 2.2.5. The fact that $$0{\lt}\xi {\lt}1$$ ensures that the series converges on the open unit disk, and it can be proven that $$F(\xi )=x$$.

The next lemma allows to compare the seemingly uncorrelated topologies on the spaces of real and of Laurent measures:

Lemma 2.2.7
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Fix a finite set $$S$$, real numbers $$0{\lt}\xi ,p'{\lt}1$$ and set $$r'=(\xi )^{p'}$$. The map $$\vartheta _\xi$$ of Definition 2.2.4 is continuous when endowing $$\mathcal L_{r'}(S)$$ with the topology defined in Definition 2.2.2 and $$\mathcal M_{p'}(S)$$ with the topology defined in Definition 2.2.1

Proof

Omitted for now, but done in Lean.

Given any finite set $$S$$, the map $$\vartheta _{\xi }$$ defines a map $$\vartheta _{\xi ,S}\colon \mathcal L_{r'}(S)\longrightarrow \mathcal M_{p'}(S)$$ by extending $$\vartheta _{\xi }$$ componentwise. For the purpose of this project, the special case where $$\xi =1/2$$ is enough, and we write $$\theta _S$$ from now on to denote $$\vartheta _{\xi ,S}$$.

Theorem 2.2.8
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Let $$0{\lt}p'{\lt}1$$ be a real number, let $$S$$ be a finite set, and let $$r'$$ denote $$\left(\frac{1}{2}\right)^{p'}$$. The sequence

$0\longrightarrow \mathcal L_{r'}(S)\longrightarrow \mathcal L_{r'}(S)\overset {\theta _S}{\longrightarrow } \mathcal M_{p'}(S)\longrightarrow 0,$

where the first map is multiplication by $$2T-1$$, is exact. In particular, the kernel of $$\theta _S$$ is principal, generated by $$2T-1$$.

Proof

The proof of surjectivity for finite $$S$$ follows immediately from Lemma 2.2.6, and the description of the kernel is straightforward. Similarly, the proof of continuity is a direct consequence of Lemma 2.2.7.