Formal Definitions
An outer measure on a set X is a function
defined on all subsets of X, that satisfies the following conditions:
- Null empty set: The empty set has zero outer measure (see also: measure zero).
- Monotonicity: For any two subsets A and B of X,
- Countable subadditivity: For any sequence {Aj} of subsets of X (pairwise disjoint or not),
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
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 integrals.
Read more about this topic: Outer Measure
Famous quotes containing the words formal and/or definitions:
“This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
—Leontine Young (20th century)
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)