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 a non-negative 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
That is, the infimum extends over all sequences of elements of C which cover E, with the convention that the infimum is infinite if no such sequence exists. 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 a non-negative extended real valued function on C which vanishes on the empty set. For each δ > 0, let
and
Obviously, φδ ≥ φδ' when δ ≤ δ' since the infimum is taken over a smaller class as δ decreases. Thus
exists (possibly infinite).
Theorem. φ0 is a metric outer measure on X.
This is the construction used in the definition of Hausdorff measures for a metric space.
Read more about this topic: Outer Measure
Famous quotes containing the words construction of, construction, outer and/or measures:
“The construction of life is at present in the power of facts far more than convictions.”
—Walter Benjamin (1892–1940)
“There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
—John Dewey (1859–1952)
“I know, it must have been my imagination, but it makes me realize how desperately alone the Earth is. Hanging in space like a speck of food floating in the ocean. Sooner or later to be swallowed up by some creature floating by.... Time will tell, Dr. Mason. We can only wait and wonder. Wonder how, wonder when.”
—Tom Graeff. Young astronomer, Teenagers from Outer Space, after just seeing the invading spaceship through his telescope, and dismissing it (1959)
“thou mayst know,
That flesh is but the glass, which holds the dust
That measures all our time;”
—George Herbert (1593–1633)