The literature on the measurement of a Euclidean set




 

ConceptDetails
Measure of a setLength, area, and volume in Euclidean space.
Basic Concepts

  • Sigma-Algebra:

    • Collection of subsets of a set X with:

    • The empty set included.

    • Complements of sets included.

    • Countable unions of sets included.



  • Measure:

    • Function μ assigning non-negative real numbers to sets in a sigma-algebra, satisfying:

    • μ(empty set) = 0.

    • Countable additivity: μ(union of disjoint sets) = sum of the measures of the sets.




Lebesgue Measure
  • Generalizes length, area, and volume.
  • In the real line:
    • Measure of interval [a, b] is b – a.
  • In the plane:
    • Measure of rectangle [a, b] x [c, d] is (b – a)(d – c).
  • In three-dimensional space:
    • Measure of box [a, b] x [c, d] x [e, f] is (b – a)(d – c)(f – e).
Properties of Lebesgue Measure
  • Translation Invariance: Measure remains the same when a set is translated.
  • Countable Additivity: Measure of a countable union of disjoint sets is the sum of their measures.