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Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."

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Measure Thoery: Chp 2. Functions and Integrals

This chapter is devoted to the definition and basic properties of the Lebesgue integral.

Measurable Functions

Let be a measurable space, and let be a subset of that belongs to . For a function the conditions - for each real number the set belongs to , - for each real number the set belongs to , - for each real number the set belongs to , and - for each real number the set belongs to ,

are equivalent. A function is measurable with respect to if it satisfies one, and hence all of the conditions.

Properties That Holds Almost Everywhere

Let be a measure space. A property of points of is said to hold -almost everywhere if the set of points in at which it fails to hold is -negligible (-a.e. or a.e.).

The Integral

Let be a measurable space. We will denote by the collection of all simple real-valued -measurable functions on and by the collection of nonnegative functions in . If and is given by (Double subscripts: use braces to clarifya_i _+ , and disjoint subsets of in ), then , the integral of with respect to , is defined to be .

Limit Theorems

The Monotone Convergence Theorem Let be a measure space, and let and be -valued -measurable functions on . Suppose that

and

hold at -almost every . Then .

Beppo Levi’s Theorem Let be a measure space, and let be an infinite series whose terms are -valued -measurable functions on . Then

Fatou’s Lemma Let be a measure space, and let be a sequence of -valued -measurable functions on . Then

Lebesgue’s Dominated Convergence Theorem Let be a measure space, let be a -valued integrable function on , and let and be -valued -measurable functions on such that

and

hold at -almost every . Then and are integrable, and .

The Riemann Integral

Theorem 2.5.4 Let be a closed bounded interval, and let be a bounded real-valued function on . Then

  • is Riemann integrable if and only if it is continuous at almost every point of , and
  • if is Riemann integrable, then is Lebesgue integrable and the Riemann and Lebesgue integrals of coincide.

Measurable Functions Again, Complex-Valued Functions, and Image Measures

References

[1] @book{cohn2013measure, title={Measure theory}, author={Cohn, Donald L}, year={2013}, publisher={Springer} }