Measure, Integration & Real Analysis, 2021a

Chapter 2
Measures
The last section of the previous chapter discusses several deficiencies of Riemann
integration. To remedy those deficiencies, in this chapter we extend the notion of the
length of an interval to a larger collection of subsets of R. This leads us to measures
and then in the next chapter to integration with respect to measures.
We begin this chapter by investigating outer measure, which looks promising but
fails to have a crucial property. That failure leads us to σ-algebras and measurable
spaces. Then we define measures in an abstract context that can be applied to settings
more general than R. Next, we construct Lebesgue measure on R as our desired
extension of the notion of the length of an interval.
Fifth-century AD Roman ceiling mosaic in what is now a UNESCO World Heritage
site in Ravenna, Italy. Giuseppe Vitali, who in 1905 proved result 2.18 in this chapter,
was born and grew up in Ravenna, where perhaps he saw this mosaic. Could the
memory of the translation-invariant feature of this mosaic have suggested to Vitali
the translation invariance that is the heart of his proof of 2.18?
CC-BY-SA Petar Milošević
Measure, Integration & Real Analysis, by Sheldon Axler
13