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Can we replace finite with countably infinite sequences in the definition of absolute continuous functions?

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Absolute Continuous functions require a finite sequence of subintervals of the domain. Can we replace this finite sequence with a countably infinite one in this definition?

Please note that an absolutely continuous function preserves null sets. To prove this result, I choose an open set G that contains a given null set N. Then eventually, I proved that the image of N has measure zero. In this proof, at some point, I chose a finite sequence from the countable sequence of intervals that covers G. Moreover, the converse is also true ie any continuous function with a bounded variation that preserves the null set is absolutely continuous. Based on my own shallow observation, I feel like we can replace this finite sequence criterion with a countably infinite one. Is this true? Could you give me an easy counter-example where my guess is false? Thank you for your time.


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