Lebesgue Measurable Functionsn and Integration

Lebesgue Measurable Function- An extended real valued  function f is Lebesgue measurable if its domain is measurable and if it satisfies one of the following five; for each real number α,
    (a) {x : f(x) < α} is measurable
          (b) {x : f(x) > α} is measurable
          (c) {x : f(x) ≤ α} is measurable
          (d) {x : f(x) ≥ α} is measurable
          (e) {x : f(x) = α} is measurable.

Almost everywhere property- If a set of points where it fails to hold is a set of measure zero.
   If f = g, almost everywhere if f and g have the same domain and m{x : f(x) ≠ g(x)} = 0.

Simple Function- A real valued function ∅ is simple function, if it is lebesgue measurable and assume only a finite number of values.

Borel measurability- A function  f is Borel measurable if for each α, the set  {x : f(x) > α} is Borel set.

Some Important Results:

• If f is an extended real valued function whose domain is measurable. Then the following statements are equivalent : For each real number α,

1. The set {x : f(x) > α} is Lebesgue measurable.
2. The set {x : f(x) < α} is Lebesgue measurable.
3. The set {x : f(x) ≤ α} is Lebesgue measurable.
4. The set {x : f(x) ≥ α} is Lebesgue measurable.

• If c is a constant and f and g two Lebesgue measurable real valued functions defined on the same domain. Then the function f + c, cf, f + g, g - f and fg are also Lebesgue measurable.
• If f is a measurable function and f = g almost everywhere, then g is measurable.

                             Lebesgue Integration

The Riemann Integral: If f is a bounded real valued function defined on the interval [a, b] and

a = x₀ < x₁ < . . . < x_n = b is a sub-division of [a, b].

                   S = Σ(x_i - x_i-1)M_i ,              ∀ i = 1, 2, . . . , n

                   s = Σ(x_i - x_i-1)m_i ,               ∀ i = 1, 2, . . . , n

where     M_i = sup f(x)   

and         m_i = inf f(x).

Step Function- For the given subdivision

a = x₀ < x₁ < . . . < x_n = b of the interval [a, b], a function Φ is a step function if

                      Φ(x) = c_i ,       x_{i -1} < x < x_{i .}

The Lebesgue Integral of a Bounded Function Over a Set of Finite Measure:

• canonical representation of simple function- If Φ is a simple function and {a₁ , . . . ,a_n} is the set of non-zero values of Φ, then,

                  Φ = Σ a_i χ_Ai  ,      where A_i = {x : Φ(x) = a_i} is called canonical representation of simple function. Here A_i are disjoint and a_i are distinct and non-zero.

• integral of simple function: If simple function Φ vanishes outside a set of finite measure, the integral of Φ is

              ∫Φ(x)dx = Σ a_i(mA_i)
when Φ has a canonical representation Φ = Σ a_i χ_Ai  and mA_i is the Lebesgue measure of A_i.
If E is any measurable set, we have
∫_EΦ = ∫Φ · χ_E

• The Lebesgue integral: If f is a bounded measurable function defined on a measurable set E with mE finite, the Lebesgue integral of f over E is

       ∫_E f(x)dx = inf ∫_E ψ . (x)dx for all simple function ψ ≥ f.

The Integral of a Non-negative Function: If f is a non-negative measurable function defined on a measurable set E, then integral of non-negative measurable function f is
                   ∫_E f = sup ∫_E h   for all h ≤ f
where h is a bounded measurable function such that
m{x : h(x) ≠ 0} < ∞.

• Integrable function: A non-negative measurable function f is integrable over the measurable set E, if

                  ∫_E f < ∞.

The General Lebesgue Integral:

• The positive and negative part of function:
          f ⁺(x) = max {f(x), 0}
          f ^-(x) = max {-f(x), 0}
  and        f = f ⁺ - f ^-
             |f| = f ⁺+ f ^-
  If f is measurable function then f is integrable over E if f ⁺and f ^- are both integrable over E, and
    ∫_E f(x)dx = ∫_E f ⁺(x)dx + ∫_E f ^-(x)dx

Convergence in Measure:

• Convergence in measure: A sequence <f_n> of measurable function is said to converge to f in measure if, given ε > 0, there is an N such that for all n ≥ N. we have
       m {x : |f(x) - f_n(x)| ≥ ε} < ε.
• cauchy sequence in measure: A sequence <f_n> of measurable function is Cauchy sequence in measure if given ε > 0, there is an N such that for all m, n ≥ N.
  we have
       m {x : |f_n(x) - f_m(x)| ≥ ε} < ε.