I am asking for a self contained proof of this assertion:

  • If $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$.

The integral in question is the Henstock-Kurzweil, (otherwise called the gauge or generalised Riemann) integral and unlike the Riemann and Lebesgue integrals, $\min(f,g)$ is not always integrable when $f,g$ both are.

This is used in Bartle's "A Modern Theory of Integration" to establish Fatou's Lemma for the gauge integral. This was proven using the fact that:

  • If $f,g$ are gauge integrable and $\left|f\right|\le g$ then $\left|f\right|$ is also gauge integrable.

Proving this however requires the Theorem:

  • If $f$ is gauge integrable then $\left|f\right|$ is gauge integrable if and only if $F(x)=\int_a^xf$ is of bounded variation.

So basically is there a self contained proof of the 1st (or even the 2nd) assertion without using the 3rd theorem (or even the notion of bounded variation)? I ask this because it seems unnatural to characterise absolute integrability and use this to deduce the convergence results (Fatou's Lemma, Dominated Convergence Theorem, Mean Convergence Theorem)


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