ormal definition of measurability of a real-valued function

A real-valued function f defined on a set X is said to be measurable if for every real number α, the set {x ∈ X : f(x) > α} is a measurable set.

Here are the steps to understand this definition:

1. We start with a real-valued function f, which means a function that takes real numbers as input and produces real numbers as output. This function is defined on a set X, which means that the inputs to the function come from the set X.

2. We consider an arbitrary real number α.

3. We then consider the set of all inputs x from X such that f(x) > α. This is a subset of X, and it depends on the choice of α.

4. We say that the function f is measurable if this set is a measurable set for every choice of α.

A measurable set, in this context, is a set that can be assigned a measure, which is a way of assigning a size or volume to sets. The precise definition of a measurable set depends on the measure space that is being used. In the context of real-valued functions, the most common measure space is the set of real numbers with the standard Lebesgue measure, in which case a set is measurable if it can be approximated by a union of intervals.