Calculus of Real and Complex Variables

6.2 The Lebesgue Integral For Nonnegative Simple Functions

To begin with, here is a useful lemma.

Proof: From the definition,

Now the Lebesgue integral for a nonnegative function has been defined, what does it do to a nonnegative simple function? Recall a nonnegative simple function is one which has finitely many nonnegative real values which it assumes on measurable sets. Thus a simple function can be written in the form

∑n s (ω ) = ciXEi (ω) i=1

where the c_i are each nonnegative, the distinct values of s.

Proof: Without loss of generality, assume 0 ≡ a₀ < a₁ ≤ a₂ ≤

≤ a_p and that μ < ∞,i > 0. Here is why. If μ = ∞, then letting a ∈, by Lemma 6.2.1, the left side would be

and so both sides are equal to ∞. Thus it can be assumed for each i,μ < ∞. Then it follows from Lemma 6.2.1 and Lemma 6.1.2,

∫ ∫ p ∫ ∞ μ ([s > λ])dλ = ap μ([s > λ])dλ = ∑ ak μ([s > λ])dλ 0 0 k=1 ak−1

∑p ∑p ∑p ∑ i ∑p = (ak − ak− 1) μ (Ei) = μ (Ei) (ak − ak− 1) = aiμ (Ei) ■ k=1 i=k i=1 k=1 i=1

Proof: Let

∑n ∑m s(ω ) = αiXAi(ω),t(ω) = βjXBj (ω) i=1 i=1

where α_i are the distinct values of s and the β_j are the distinct values of t. Clearly as + bt is a nonnegative simple function because it has finitely many values on measurable sets. In fact,

∑m ∑n (as+ bt)(ω) = (aαi + bβj)XAi∩Bj(ω ) j=1 i=1

where the sets A_i ∩ B_j are disjoint and measurable. By Lemma 6.2.2,