Motivation for Defining Measurable Functions

Question

I have just started Measure Theory from Bartle, The Elements of Integration and Lebesgue Measure, where he defines Measurable Functions as:

What is the motivation for this? How is it related/equivalent to the more standard way of defining measurable functions by considering elements as pre-images?

Answer 1

If we consider the measurable space $(X , \mathscr{F})$, then a function $f: X \rightarrow X$ is said to be measurable if the preimage $f^{-1}(A) \in \mathscr{F}$ for all sets $A \in \mathscr{F}$. Note that I have changed the bold X in your screenshot to the more common notation of $\mathscr{F}$ - this is to avoid confusion between $X$ and X (which might be creating some unnecessary confusion).

Expanding out the definition of a preimage tells us that this is equivalent to requiring that :

$$ \{ x \in X : f(x) \in A \space \text{and} \space A \in \mathscr{F} \} \in \mathscr{F}$$

Note that if we let $A$ be the interval $(\alpha, + \infty)$ and substitute this into the above definition, then we have the result in the definition you have posted. Here the function $f$ is said to be $\mathscr{F}$-measurable.

Answer 2

If $X$ a set and $\mathcal X$ is a $\sigma$-algebra on it then for a function $f:X\to\mathbb R$ the following statements are equivalent:

• $f$ is Borel-measurable
• For every $B\in\mathcal B(\mathbb R)$ we have $f^{-1}(B)\in\mathcal X$
• For every $\alpha\in\mathbb R$ we have $\{x\in X\mid f(x)>\alpha\}\in\mathcal X$

(where $\mathcal B(\mathbb R)$ denotes the Borel $\sigma$-algebra on $\mathbb R$ i.e. the smallest $\sigma$-algebra that contains all open subsets of $\mathbb R$)

Note that here $\{x\in X\mid f(x)>\alpha\}=f^{-1}((\alpha,\infty))$ while $(\alpha,\infty)\in\mathcal B(\mathbb R)$ so actually the condition under the second bullet implies the condition under the third bullet directly.

Conversely the condition under the third bullet is enough to prove that the condition under the second bullet is satisfied. This because it can be proved that:$$\sigma(\{(\alpha,\infty)\mid\alpha\in\mathbb R\})=\mathcal B(\mathbb R)$$ and secondly in general:$$f^{-1}(\sigma(\mathcal V))=\sigma(f^{-1}(\mathcal V))$$for every collection $\mathcal V\subseteq\mathcal P(\mathbb R)$ (so for instance for $\mathcal V:=\{(\alpha,\infty)\mid\alpha\in\mathbb R\}$).

Which one (second or third bullet) to use as definition is a matter of choice.