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math105-s22:notes:lecture_9

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Lecture 9

We will cover Tao's 7.5 and 8.1 today. Here we will use Tao's definition of measurable set, and Lebesgue integration, which a priori is not the same as Pugh's.

Tao 7.5: Measurable function

Let ΩRm\Omega \In \R^m be measurable, and f:ΩRmf: \Omega \to \R^m be a function. If for all open sets VRmV \In \R^m, we have f1(V)f^{-1}(V) being measurable, then ff is called a measurable function.

If f:ΩRmf: \Omega \to \R^m is continuous, then ff is measurable. Indeed, if f1(V)f^{-1}(V) is open in Ω\Omega, then f1(V)f^{-1}(V) is an intersection of open subset URmU \In \R^m and Ω\Omega (recall the definition of topology on Ω\Omega), an intersection of two measurable sets.

Instead of checking on all open sets VRmV \In \R^m, we can just check for all open boxes in Rm\R^m. Since any open can be written as a countable union of open boxes.

A measurable function ff, post compose with a continuous function gg is still measurable. Since (gf)1(open)=f1(g1(open))=f1(open)=measurable(g \circ f)^{-1}(open) = f^{-1} (g^{-1}(open)) = f^{-1}(open) = measurable

Lemma: f:ΩRf: \Omega \to \R is measurable if and only if for all aRa \in \R, f1((a,))f^{-1}( (a, \infty)) is measurable.
Proof: every open set in R\R is a countable union of open interval, hence suffice to show that all open intervals (a,b)(a,b) has pre-image being measurable. We can easily show that f1((a,b])f^{-1}((a, b]) is measurable for all a<ba<b, and we can use countable operations to approximate open interval by half-open-half-closed ones, (a,b)=n(a,b1/n](a,b) = \cup_{n} (a, b-1/n].

8.1 Simple function

Simple functions are measurable functions f:ΩRf: \Omega \to \R, which takes value in a finite subset of R\R.

Simple functions forms a vector space (i.e., closed under addition and scalar multiplication), and can be written as a finite linear combination of characteristic functions χE\chi_E.

The important thing is that, any non-negative measurable function ff admits a sequence of simple functions fnf_n, non-negative, and fnfn+1f_n \leq f_{n+1}, such that fnff_n \to f pointwise.

We then define

math105-s22/notes/lecture_9.1644900910.txt.gz · Last modified: 2022/02/14 20:55 by pzhou