Let (X,S,µ) be a measure space. [0;+ 1 ] be measurable. It is useful to compare the definition of a σ-algebra with that of a topology in Definition 1.1. A measurable space (X,A) is a non-empty set Xequipped with a σ-algebra A on X. Proposition 4: Given any non-negative measurable function on , there0 H is a sequence of simple functions such that .00Å088 8Ä_ Proof: Let 0ÐBÑœ 0ÐBÑ−Ò ß Ñ 0ÐBÑ 8 8 80ÐBÑ 8 777 " œ##888 if and if 2 . Definition 1.5. 4.3. ... Find an example of a non-measurable function ffor which jfjis measurable. 1. If f: !Sand g: S!T are measurable, then g(f) : !T is measurable.

function can be written as P k a k 1 S k in many ways, the integral will always be the same .24 Non-negative functions : Let f : E ! A measurable function f from X to IR is said to be integrable over a set E∈ Sif both R E f+ dµand R E f− dµare finite. If f is measurable, so are both f+ and f−.

Theorem 1.1. Theorem. lim Theorem 4.3 (Monotone Convergence). Integrals of Measurable Nonnegative Function 5 Monotone Convergence Theorem.

.,an). Proof. .,n are intervals in R, the graph of the simple function f looks like a collection of steps (of heights a1,. Suppose Ω is a nonempty set and A is a σ-field on it.

If αand βare positive real numbers, then Z X (αψ+βϕ)dµ= α Z

6 Fact.4 A function f : E !R is simple if and only if there exists a 1;a Let f be a function from a measurable space (X,S) to IR.

Integral is Additive for All Non-negative Measurable Functions Interchanging Summation and Integration Fatou's Lemma : 5: Integral of Complex Functions Dominated Convergence Theorem Sets of Measure Zero Completion of a Sigma-algebra : 6: Lebesgue Measure on R^n Measure of Special Rectangles Measure of Special Polygons A function f: E!R which is measurable and takes only nitely many values is called a simple function.

Let fn be an increasing se­ quence of non-negative measurable (extended) functions, then f (x) = n ∗ fn(x) is measurable and (4.5) f dµ = lim fndµ E n ∗ E A function f from X to IR is called measurable if, for each a ∈ IR, {x ∈ X : f(x) > a} is a measurable set. Integration of Nonnegative Measurable Functions 2 Proposition 18.8. 18.2. First, the complement of a measurable set is measurable, but the complement of an open set is not, in general,

If fand gare measurable real-valued functions, then f+ gand fgare 34 3.

If f is an extended real-valued measurable function and ais a constant, then af is measurable. Every measurable function is the difference of two nonnegative measurable functions.

The Lebesgue integral is very easy to define for non-negative There are two significant differences. If s = P n i=1 c iχ E i … on E, then lim n→∞ Z E fn Z E lim n→∞ fn E is a measurable function.