Lebesgue Measure

Measure:

• Length of an interval: The length of an interval I is the difference of end points of the interval.
• Measure of a set: Let M    be a collection of sets of real numbers and E ∈ M   . Then non-negative extended real number mE is called the measure of E. if m satisfies:
      (a) mE is defined for each set E of real numbers. i.e.,
            M   = P(R), the power set of sets of real number.
      (b) For an interval I, mI = l(I).
      (c) If <E_n> is a sequence of disjoint sets
            (for which m is defined) in M   .
               m(∪E_n) = ΣmE_{n.
      (d) m is translation invariant, i.e.,  if E is the set on which m is defined, and
                   E + y = {x + y : x ∈ E} then m(E + y) = mE.}
• _{Countable additive measure: let M   be a σ-algebra of sets of real numbers and} E ∈ M   . Then non-negative extended real number mE is countable additive measure, if m(∪E_n) = ΣmE_n  for each sequence <E_n> of disjoint sets in M   .
• Countable sub-additive measure:  _{let M   be a σ-algebra of sets of real numbers and} E ∈ M   . Then non-negative extended real number mE is countable sub-additive measure, if m(∪E_n) ≤ ΣmE_n  for each sequence <E_n> of disjoint sets in M   .

Lebesgue Outer Measure: Let A be a set of real numbers, {I_n} be the countable collection of open intervals that covers A, i.e., A ⊂ ∪I_n. Then Lebesgue outer measure m^*A of A is

           m^*A = Inf Σl(I_n)  where A ⊂ ∪I_n

• Some Important Results:
  (1) m^*∅ = 0.
  (2) A ⊂ B ⇒ m^*A ≤ m^*B.
  (3) For singleton set {x}, m^*{x} = 0.
  (4) The Lebesgue outer measure of an interval is its length.
  (5) If {A_n} is a countable collection of sets of real number. Then m^*(∪A_n) ≤ m^*A_n
  (6) If A is countable then m^*A = 0.
  (7) The set [0, 1] is not countable.

Lebesgue Measurable Sets and Lebesguue Measure:

• Lebesgue measurable sets- A set E is Lebesgue measurable if for each set A we have m* A = m* (A∩E) + m* (A∩E^c).
• Lebesgue measure- If E is a Lebesgue measurable set, the Lebesgue measure mE is the Lebesgue outer measure of E.
• Some important Results:
  (1) If  m^*E = 0, then E is Lebesgue measurable.
  (2) If E₁ and E₂ are Lebesgue measurable, so E₁ ∪ E₂.
  (3) The family M   of Lebesgue measurable sets is an algebra of sets.
  (4) The collection M   of Lebesgue measurable sets is a σ-algebra.
  (5) Every set with Lebesgue outer measure zero is Lebesgue measurable.
  (6) The interval (a, ∞) is Lebesgue measurable.
  (7) Every Boral set is Lebesgue measurable.
  (8) Each open set and closed set is Lebesgue measurable.
  (9) If <E_i> is a sequence of Lebesgue measurable set. Then for Lebesgue measure m(∪E_i) ≤ ΣmE_i
  If the sets E_i are pairwise disjoint the for Lebesgue measure m(∪E_i) = ΣmE_i
  (10) Let <E_n> be an infinite decreasing sequence of Lebesgue measurable sets, i.e., E_n+1 ⊂ E_n for each n. Let Lebesgue measure mE, is finite. Then
                           m(∪E_i) = lim  mE_n for all n = 1, . . . , ∞.

Littlewood's Three Principles:
(a) Every (measurable) set is nearly a finite union of intervals.
(b) Every (measurable) set is nearly continuous.
(c) Every convergent sequence of (measurable) function is nearly uniform convergent.

Jacobian Determinant:
If f = (f₁ , f_{2 , … ,} f_n) and x= (x₁ , x_{2 , … ,} x_n) the Jacobian matrix is Df (x) = [D_if_i(x)] on n × n matrix, and Jacobian determinant is
                J_f(x) = det. Df (x) = det [D_if_i(x)].