I've just started self-studying measure theory and the book I'm using has the following exercise.

Let $J$ be an interval in $\mathbb{R}$ and $\mathcal{S}_J = \{S \subseteq J\ | S\ \text{is bounded} \}$. Also, let $\mu: \mathcal{S}_J \rightarrow \mathbb{R}$ be a finitely additive measure. Show that there exists a function $F:J\rightarrow \mathbb{R}$ such that

$\mu([a,b)) = F(b)-F(a)\, \quad$ if $a<b$ are in $J$.

I wonder if I'm missing something, but wouldn't $F(x)=x$ already do the trick?



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