ABSTRACT
In this chapter we consider functions that are measurable with respect to a given σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math2_1.tif"/> -field F $ \boldsymbol{{ \fancyscript {F}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math2_2.tif"/> , that is, functions f for which (in the real-valued case) the sets { x ∈ X : f ( x ) ∈ ( a , b ) } $ \{x \in X: f(x) \in (a,b)\} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math2_3.tif"/> are F $ \boldsymbol{{ \fancyscript {F}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math2_4.tif"/> -measurable. As we shall see, such functions are natural candidates for integration with respect to Lebesgue measure. We begin with the more general notion of measurable transformation.
