Blueprint for the Liquid Tensor Experiment: Spaces of Measures

2.2 Spaces of Measures

Definition 2.2.1 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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 ✓

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.