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:
“There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
—Sara Lawrence Lightfoot (20th century)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)