# Construction of Lebesgue measure using two different approaches.

I'm going to study in a possibly serious way the construction of Lebesgue measure. What are the advantages/disadvantages of the following two approaches in order to build the Lebesgue measure?

1. Constructing Lebesgue measure using Carathéodory theorem and outer measure (see Folland);

2. Constructing Lebesgue measure through Riesz representation theorem (see Rudin).

I have heard that the first one is more general and technically easy. Is it true?

I have already studied in a soft way the construction of Lebesgue measure on $$\mathbb{R}^n$$, by defining Lebesgue measurable sets to be the ones that can be well approximated by open sets from the outside and by closed sets from the inside. Is this the most direct way to proceed? Again, which one of the previous two approaches is more similar to this last construction?

Again, I'm also interested in Lebesgue-Stieltjes measure construction. Maybe with the first approach (Carathéodory) I can obtain both Lebesgue and Lebesgue-Stieltjes measures at the same time?

Thank you!

• If you are going to study in a serious way the construction of Lebesgue measure, you must learn both ways. So the question is actually which one to study first. In general, you should start by the approach "using Carathéodory theorem and outer measure" first and then study the approach "through Riesz representation theorem", unless you have a specific reason / motivation to do in the reversed order. Commented Jan 24, 2021 at 13:21