Question on a pre-measure defined by Folland's real analysis book

My simple question might suggest I'm missing something quite basic. I, therefore, apologize in advance.

In Prop 1.15, Folland defines a pre-measure on h-intervals (left open right closed intervals).

1.15 Proposition. Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be increasing and right continuous. If $\left(a_j, b_j\right]$ $(j=1, \ldots, n)$ are disjoint $h$-intervals, let

$$\mu_0\left(\bigcup_1^n\left(a_j, b_j\right]\right)=\sum_1^n\left[F\left(b_j\right)-F\left(a_j\right)\right],$$

and let $\mu_0(\emptyset)=0$. Then $\mu_0$ is a premeasure on the algebra $\mathcal{A}$.


My problem is the following. Folland also includes in the h-intervals Algebra sets of the following form $(a,\infty)$
How are these defined under his definition of $\mu_0$?

Measure Theory
Dannyboy990 Dannyboy990
31
Report

Answer

Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer

1 Attachment

Erdos Erdos
4.8K
  • Erdos Erdos
    0

    Please leave a comment if you need any clarifications.

    • Dannyboy990 Dannyboy990
      0

      Why should it be closed under countabe union? The pre-measure is defined on an Algebra which is closed under finite unions.

    • Erdos Erdos
      0

      If you read the bottom of page 30 in Folland, in defining a premature the goal is to extend measure from algebras to \sigma-algebras. So by definition a pre-measure is defined on the smallest \sigma-algebra that contain the given algebra. So you need the sets to be closed under countable union, and not just finite union. I attached page 30 of Folland.

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.