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Let $f:[a,b]\to \mathbb{R}$ so that $f(x)=0$ almost everywhere in $[a,b]$. Prove that $f$ is gauge integrable and $\int_a^bf=0$.

How can this be proven using the following definition of measure: $\mu(A)=\int_{-\infty}^{\infty}\chi_A$ when of course $\chi_A$ is integrable or the integral is infinite? I have established the basic properties of measures (countable additivity etc.) and can use the MCT, Fatou's Lemma, DCT for the gauge integral. Almost all proofs I have seen either use measurability of $f$ or the characterisation of measure $0$ via open interval covers. I would like to see an alternate approach

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