Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.