Is it true that the infimum of all piecewise constant functions that majorize a function $f$ is $f$ itself? If yes, how do I show it formally?
Definition 1. Let $I$ be bounded interval. Let $f:I \to \mathbb{R}$ be bounded function. We call $f$ piecewise constant on $I$ iff there exists some partition $P$ of $I$, such that $f$ is constant on each $J\in P$.
Definition 2. .... We say, f majorizes g, iff $\forall x \in I: f(x) \geq g(x)$.
You have not put any condition on $f$ except boundedness. If $f$ is assumed to be continuous then the result is true. In general it is false. For example, if $f(0)=-1$ and $f(x)=0$ for $x \neq 0$ then, for any pieceiwise constant function $g$ with $g \geq f$ we have $g(0) \geq 0$ so the infimum of all such functions is not equal to $f$.