(H. Priestley - Introduction to Integration - Exercise 4.3)
Define the class $\mathbb L$ of integrable functions for which the following $Basic Properties$ hold:
(1) Building Block: $ \forall a,b \in \mathbb R $ the characteristic function is integrable and $\int \chi_{[a,b]} := (b-a)$
(2) Linearity: if $f,g$ are integrable and $\lambda \in \mathbb R$ then $f+ \lambda g$ is integrable and $\int (f+\lambda g)=\int f + \lambda \int g$
(3) Positivity: if $f$ is integrable then $f(x) \ge 0 \ \forall x \implies \int f \ge 0$
(4) Modulus property: if $f$ is integrable then so is $|f|$
Prove that the class of step functions is the minimal set of functions in $\mathbb L$ for which these properties hold.
Normally I give an attempt at the answer but in this case, I have no clue how to even start ... I am not even sure I understand what a $minimal \ set$ of functions is.
Say $S$ is the class of all step functions. Saying that $S$ is the minimal class where properties 1-4 hold means two things:
Properties 1-4 hold for the class $S$.
If $C$ is a class of functions for which 1-4 hold then $S\subset C$.
So that says you have to do two things. First, show that $S$ satisfies properties 1-4. Second, assume that $C$ satisfies 1-4, and then use those properties of $C$ to show that $C$ contains every step function.