Outer Measure

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than mere intervals or open balls in R³. One might expect to define a generalized measuring function φ that fulfils the following three requirements:

# Any interval of reals ''b'' has measure ''b'' − ''a''
# The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
# (''A''_''j'')_''j'' of pairwise Disjoint Subsets of ''X''

::

arphi\left(\bigcup_{i=1}^\infty A_i
ight) = \sum_{i=1}^\infty arphi(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 pick out a class of subsets (to be called ''measurable'') in such a way as to satisfy the countable additivity property.

FORMAL DEFINITIONS

An outer measure is a function defined on all subsets of a set ''X''
:

arphi: 2^X 
ightarrow \infty

such that

The Empty Set has zero outer measure ( Measure Zero ).

arphi(arnothing) = 0

Monotonicity

A \subseteq B \Rightarrow arphi(A) \leq arphi(B)

Countable sub-additivity: for any sequence {''A''_''j''}_''j'' of subsets of ''X'' (pairwise disjoint or not)

arphi\left(\bigcup_{j=1}^\infty A_j
ight) \leq \sum_{j=1}^\infty arphi(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''

:

arphi(A) = arphi(A \cap E) + arphi(A \setminus E).

Theorem. 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 Integral s.

OUTER MEASURE AND TOPOLOGY

Suppose (''X'', ''d'') is a Metric Space and φ an outer measure on ''X''. If φ has the property that

:

arphi(E \cup F) = arphi(E) + arphi(F)

whenever

:

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

then φ is called a ''metric outer measure''.

Theorem. If φ is a metric outer measure on ''X'', then every Borel subset of ''X'' is φ-measurable. (The Borel Set s of ''X'' are the elements of the smallest σ-algebra generated by the open sets.)

CONSTRUCTION OF OUTER MEASURES

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 family of subsets of ''X'' which contains the empty set and ''p'' an extended real valued function on ''C'' which vanishes on the empty set.

Theorem. Suppose the family ''C'' and the function ''p'' are as above and define

:

arphi(E) = \inf \left\{ \sum_{i=1}^\infty p(A_i)
ight\}

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 family of subsets of ''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

:

arphi_\delta(E) = \inf \left\{ \sum_{i=1}^\infty p(A_i)
ight\}

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 
ightarrow 0} arphi_\delta(E) = arphi_0(E) \in \infty

exists.

Theorem. φ₀ is a metric outer measure on ''X''.

This is the construction used in the definition of Hausdorff Measure s for a metric space.

REFERENCES

P. Halmos , ''Measure theory'', D. van Nostrand and Co., 1950

M. E. Munroe, ''Introduction to Measure and Integration'', Addison Wesley, 1953

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Information About

Outer Measure