外测度

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外測度是個實數集合的函數,並符合一些額外條件。 Carathéodory 所創立的理論為測度集合理論建立基礎。 Carathéodory 在外測度的工作對於測度集合論很有用。and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension.

長度，面積及體積的歸納出來的測度對很多抽象不規則的集合很有用的。我們可以定義φ作為測度函數,其滿足以下3個條件:

任意實數間距 [a, b] 等於 b − a
測度函數 φ 是非負實函數於R的所有子集合定義下來
可數相加律, 設定X 的對偶分離子集的任意序列{A_j}_j

$\varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i)$

It turns out the second and third requirements together for all sets are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of X is to suitably pick out a class of subsets (to be called measurable) in such a way that fulfils the countably additivity property.

[编辑] 定義

外測度是個X的冪集合(Power set)映射到 $[0, \infty]$ 的測度

$\varphi: 2^X \rightarrow [0, \infty]$

固此

空集有零外測度(測度零).

$\varphi(\varnothing) = 0$

一貫性

$A \subseteq B \Rightarrow \varphi(A) \leq \varphi(B)$

可數相加律: 設定X 的(無論是對偶分離與否)子集的任意序列{A_j}_j

$\varphi\left(\bigcup_{j=1}^\infty A_j\right) \leq \sum_{j=1}^\infty \varphi(A_j)$

This allows us to define the concept of measurability as follows: a subset E of X is φ-measurable (or Carathéodory-measurable by φ) iff for every subset A of X

$\varphi(A) = \varphi(A \cap E) + \varphi(A \setminus E).$

定理. The φ-measurable sets form a σ-algebra and φ restricted to the measurable sets is a countably additive complete measure.

For a proof of this theorem see the Halmos reference, section 11.

This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure that is so important for measure theory and the theory of integrals.

[编辑] 外測度及拓墣學

假設 (X, d) 是一個度量空間且 φ 是一個在X之上的外測度。若φ 有以下性質

$\varphi(E \cup F) = \varphi(E) + \varphi(F)$

每當

$d(E,F) = \inf\{d(x,y): x \in E, y \in F\} > 0,$

則 φ 稱為一個 metric outer measure. X的Borel sets 是 the elements of the smallest σ-algebra generated by the open sets.

定理. 若 φ 是一個 metric outer measure on X, 則每一個 X 的Borel subset是 φ-可測.

[编辑] 外測度的建立

There are several procedures for constructing outer measures on a set. The classic Munroe reference below describes two particularly useful ones which are referred to as Method I and Method II.

Let X be a set, C a subset of 2^X which contains the empty set and p an extended real valued function on C which vanishes on the empty set.

Theorem. 假設以上的類別C及函數p的定義如下：

$\varphi(E) = \inf \left\{ \sum_{i=1}^\infty p(A_i)\right\}$

where the infimum extends over all sequences {A_i}_i of elements of C which cover E (with the convention that if no such sequence exists, then the infimum is infinite). Then φ is an outer measure on X.

The second technique is more suitable for constructing outer measures on metric spaces, since it yields metric outer measures.

Suppose (X,d) is a metric space. As above C is a subset of 2^X which contains the empty set and p an extended real valued function on C which vanishes on the empty set. For each δ > 0, let

$C_\delta= \{A \in C: \operatorname{diam}(A) \leq \delta\}$

and

$\varphi_\delta(E) = \inf \left\{ \sum_{i=1}^\infty p(A_i)\right\}$

where the infimum extends over all sequences {A_i}_i of elements of C_δ which cover E. Obviously, φ_δ ≥ φ_δ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus

$\lim_{\delta \rightarrow 0} \varphi_\delta(E) = \varphi_0(E) \in [0, \infty]$