Dependencies

For any measure \(\mu \) on a local \(X\), the caratheodory extension is:

A sublocal \(X\) of \(E\) is called closed if \(X = E \setminus U\) for some open sublocal \(U\) of \(E\).

The complement of an open sublocal \(U\) of \(X\) is the sublocal \(X \setminus U\). (Leroy p. 12)

Let \(E\) be a space with \(U, H \in O(E)\). We donote by \(e_U\) the largest \(W \in O(E)\) such that \(W \cap U \subset H\). We verify that \(e_U\) is the nucleus of a subspace, which we will temporarily denote by \([U]\).

An embedding is a morphism that satisfies the conditions of 3

For every continuous Function \(f : X \rightarrow Y\) between topological Spaces, there exists a pair of functors \((f^*,f_*)\).

1. \(Int X\) is the largest open sublocal contained in \(X\)

2. \(Ext X\) is the largest open sublocal contained in \(E \setminus X\)

3. \(\bar{X}\) is the smallest closed sublocal containing \(X\)

4. \(\partial X = \bar{X} \cap (E - Int X)\)

A measure on a local \(X\) is a map \(\mu : O(X) \to [0,\infty )\) such that:

1. \(\mu (\emptyset ) = 0\)

2. \(U \subset V \implies \mu (U) \le \mu (V)\)

3. \(\mu (U \cup V) = \mu (U) + \mu (V) - \mu (V \cap V)\)

4. For any increasingly filtered family \(V_i\) of open sublocals of \(X\), we have:

   \[ \mu (\bigcup V_i) = \sup _i \mu (V_i) \]

   this means: For all \(i\) and \(j\) there exists a \(k\) such that \(V_i \cup V_j \subset V_k\) bzw. \(V_i \subset V_k\) and \(V_j \subset V_k\).

(Leroy III.1.)

A neighborhood of a sublocal \(A\) of \(X\) is an open sublocal \(V\) of \(X\) such that \(A \le V\).

A nucleus is a map \(e : O(E) \rightarrow O(E)\) with the following three properties:

1. \(e\) is idempotent

2. \(U \le e U\)

3. \(e(U \cap V) = e(U) \cap e(V)\)

For two nuclei \(e\) and \(f\) on \(O(E)\), we say that \(e \le f\) if \(e(U) \le f(U)\) for all \(U \in O(E)\). This relation is a partial order.

For any \(U \in O(E)\), the sublocal \([U]\) is called an open sublocal of \(E\).

A local is regular, if for all open sublocals \(U\) of \(E\), the open sublocals \(V\) such that \(V\bar\subset U\) recover \(U\).

(Leroy CH 3) A sublocal \(Y \subset X\) is defined by a nucleus \(e_Y: O(X) \rightarrow O(X)\), such that \(O(Y) = Im(e_Y) = \{ U \in O(X) | e_Y(U) = U\} \). The corresponding embedding is \(i_X : O(Y) \rightarrow O(X)\). \(i^*_X(V) = e_X(V)\), \((i_X)_*(U) = U\) And every nucleus \(e\) on \(O(X)\) defines a sublocal \(Y\) of \(X\) by \(O(Y) = Im(e)\)

(Stimmt das?)(Leroy Ch 3) \(X \subset Y\) if \(e_Y(u) \le e_X(u)\) for all \(u\). This means that the Sublocals are a dual order to the nuclei.

Let \((X_i)_i\) be a family of sublocal of \(E\) and \((e_i)_i\) the corresponding nuclei. For all \(V \in O(E)\), the intersection \(\bigcap X_i\) is the Union of all Nuclei \(w\) such that \(w \le x_i \) for all \(x_i \in X_i \)

(Leroy CH 1.4) Let \((X_i)_i\) be a family of sublocals of \(E\) and \((e_i)_i\) the corresponding nuclei. For all \(V \in O(E)\), let \(e(V)\) be the union of all \(W \in O(E)\) which are contained in all \(e_i(V)\).

The Caratheodory extension is subaddtive:

For any family \(X_i\) of closed sublocals of \(E\), the intersection \(\bigcap X_i\) is closed (it can be computed by taking the complement of the union of the complements).

(Leroy Lemme 3.6) For any measure on a local \(X\) and a decreasing family \((V_i)_{i\in I}\) of open sublocals, the caratheodory extension fulfills: \(\mu (\inf _{i\in I} V_i) = \inf _{i\in I} \mu (V_i)\).

(Leroy lemme 3.1) The caratheodory extension of a measure on a local commutes with unions of increasing families.

The complement is injective.

The map \(e_U\) is a nucleus.

(Leroy Lemme 1) The following arguments are equivalent:

1. \(f^*\) is surjective

2. \(f_*\) is injective

3. \(f^{*}f_* = 1_{O(X)})\)

\(f^*\) is the right adjoint to \(f_*\)

(Leroy Lemme 3.3) For any open sublocal \(U\) of a local \(X\), the caratheodory extension of a measure on \(X\) satisfies

The caratheodory extension is monotonic i.e.

(Leroy Lemme 3) Let \(e : O(E) \rightarrow O(E)\) be monotonic. The following are equivalent:

1. \(e\) is a nucleus

2. There is a locale \(X\) and a morphism \(f: X \rightarrow E\) such that \(e = f_*f^*\).

3. Then there is a locale \(X\) and a embedding \(f: X \rightarrow E\) such that \(e = f_*f^*\).

The Nuclei (and therefore the sublocals) form a complete lattice.

The Nuclei form a frame.

The Nuclei form a Heyting Algebra.

For a set \(S\) of nuclei, the intersection \(\bigcap S\) can be computed by \(\bigcap S(a) = \bigcap \{ j(a) | j \in S \} \). This function satisfies the properties of a nucleus and of an infimum.
Quelle: StoneSpaces S.51

For any open sublocal \(V\) of \(E\) and any sublocal \(X\) of \(E\), we have:

And thereby:

For any open sublocal \(V\) of \(E\) and any sublocal \(X\) of \(E\), we have:

1. \(\bar{X} = E \setminus Ext(X)\)

2. \(\partial X = E \setminus (Int X \cup Ext X)\)

3. \(Int X \cup \partial X = \bar X\)

4. \(Ext X \cup \partial X = E \setminus Int X\)

(Leroy lemme 3.2) In a regular local, any sublocal is regular, meaning that it is the intersection of all open neighborhoods.

(Leroy Lemm 3.4) For any open sublocal \(U\) and any sublocal \(A\) of a local \(E\), the caratheodory extension of a measure on \(X\) satisfies

(Leroy Lemm 3.5) For a increasing family \(V_{\alpha }\) of open sublocals of \(E\) and any sublocal \(A\), we have:

The Restriction of a Measure to any open Sublocal is a Measure.

Let \(A\) be a sublocale of \(E\) with the embedding \(i : A \rightarrow E\). The restriction of a measure \(\mu \) on \(E\) to \(A\) is a measure on \(A\):

(Leroy Lemma 6,7)

1. For all subspaces \(X\) of \(E\) and any \(U \in O(E)\):

   \[ X \subset [U] \iff e_X(U) = 1_E \]

2. For all \(U, V \in O(E)\), we have:

   \[ [U \cap V] = [U] \cap [V] \]
   \[ e_{U \cap V} = e_Ue_V=e_Ve_U \]
   \[ U \subset V \iff [U] \subset [V] \]

3. For all families \(V_i\) of elements of \(O(E)\), we have:

   \[ \cup _i[V_i] = [\cup _iV_i] \]

(Leroy CH 4) Let \(X_i\) be a family of subframes of \(E\) and \(e_i\) be the corresponding nuclei. For every \(V \in O(E)\), let \(e(V)\) be the union of all \(W \in O(E)\) which are contained in every \(e_i(V)\). Then

1. \(e\) is the corresponding nucleus of a sublocale \(X\) of \(E\)

2. a sublocale \(Z\) of \(E\) contains \(x\) if and only if it contains all \(X_i\). \(X\) is thus called the union of \(X_i\) denoted by \(\bigcup _i X_i\)

(Leroy lemme 3.7 et principal) For any measure on a local \(X\), the caratheodory extension is regular \(\mu (\inf _{i\in I} A_i) = \inf {i\in I} \mu (A_i)\). For decreasing families \((A_i)_{i\in I}\)

A complete Lattice is a Frame if and only if it as a Heyting Algebra.

(Proposition 3.3.1) For any measure on a local \(X\), the caratheodory extension is reductive, i.e. for all \(A \le X\) the set \(\{ A' \subset A, \mu (A') = \mu (A)\} \) has a minimal element.

(Leroy theorem 3.3.1) For any measure on a local \(X\), the caratheodory extension is strictly additive, i.e. \(\mu (A \cup B) = \mu (A) + \mu (B) - \mu (A \cap B)\).

For any measure on a local \(X\), the caratheodory extension is

1. strictly additiv, i.e. \(\mu (A \cup B) = \mu (A) + \mu (B) - \mu (A \cap B)\) for all \(A,B\in X\),

2. commutes with inf \(\mu (\inf _{i\in \mathbb {N}} A_i) = \inf _{i\in \mathbb {N}} \mu (A_i)\) for a familiy \((A_i)_{i\in \mathbb {N}}\) of elements \(A_i\in X\),

3. reductive, i.e. for all \(A \le X\) the set \(\{ A' \subset A, \mu (A') = \mu (A)\} \) has a minimal element.