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

Measure Thoery: Chp 3. Convergence

In this chapter we look in some detail at the convergence of sequences of functions.

Modes of Convergence

Converge in measure Let be a measure space, and let and be real-valued -measurable functions on . The sequence converges to in measure if

holds for each positive .

Converge almost everywhere The sequence converges to almost everywhere if holds at -almost every point in .

Proposition 3.1.2 Let be a measure space, and let and be real-valued -measurable functions on . If is finite and if converges to almost everywhere, then converges to in measure.

Proposition 3.1.3 Let be a measure space, and let and be real-valued -measurable functions on . If converges to in measure, then there is a subsequence of that converges to almost everywhere.

Proposition 3.1.4 (Egoroff's Theorem) Let be a measure space, and let and be real-valued -measurable functions on . If is finite and if converges to almost everywhere, then for each positive number there is a subset of that belongs to , satisfies , and is such that converges to uniformly on .

It follows from this remark and Egoroff's theorem that on a finite measure space almost everywhere convergence is equivalent to almost uniform convergence.

Converge in Mean Suppose that is a measure space and that and belong to . Then converges to in mean if

Proposition 3.1.5 Let is a measure space and that and belong to . If converges to in mean, then converges to f in measure.

Proposition 3.1.6 Let is a measure space and that and belong to . If converges to almost everywhere or in measure, and if there is a nonnegative extended real-valued integrable function such that

hold almost everywhere, then converges to in mean.

Normed Spaces

Let be a vector space over (or over ). A norm on is a function that satisfies - - if and only if - , and -

for each and in and each in (or in ). A normed vector space is a vector space together with a norm.

A metric on a set is a function that satisfies - , - if and only if , - , and -

for all and in .

Let be a metric on a set . Then a subset of is dense in if for each and each positive there is an element of that satisfies .

A sequence of elements of is a Cauchy sequence if for each positive number there is a positive integer such that holds whenever and . If every Cauchy sequence in converges to a point in , then is called complete. A normed linear space that is complete (with respect to the metric induced by its norm) is called a Banach space.

Definition of and

Let be a measure space, and let satisfy ( need not be an integer). Then is the set of all -measurable functions such that is integrable. We define by

A subset of is locally -null (or simply locally null) if for each set that belongs to and satisfies the set is -null.

Lemma 3.3.1 Let satisfy , let be defined by , and let and be nonnegative real numbers. Then

Proposition 3.3.2 (Holder's Inequality) Let be a measure space, and let and satisfy , and . If and , then belongs to and satisfies .

Proposition 3.3.3 (Minkownski's Inequality) Let be a measure space, and let satisfy . If and belongs to , then belongs to and .

Properties of and

Theorem 3.4.1 Let be a measure space, and let satisfy . Then is complete under the norm .

Dual Spaces

Proposition 3.5.1 Let and be normed linear spaces, and let be a linear operator. Then is continuous if and only if there is a nonnegative number such that

holds for each in . (Operator is called bounded.)

Dual Space Let be a normed linear space. A linear functional on is a linear operator on whose values lie in . We will be particularly concerned with the bounded, that is, continuous, linear functionals on . It is easy to check that the set of all continuous linear functionals on is a subspace of the vector space of all linear functionals on ; this subspace is called the dual space (or conjugate space) of and is denoted by . The space is sometimes called the topological dual space of V in order to distinguish it from the space of all linear functionals on V (which is then called the algebraic dual space of V).

References

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