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Recently I have encountered with this task in my calculus book. Since that I can't get it out of my mind.

Task: Let E be measurable subset of $[0,1]$ with positive measure. Prove that $f(t) = \lambda(E \cap [0,t])$ is continuous on $[0,1]$ ($\lambda$ is Lebesgue measure)

I would appreciate any help solving this task.

Mittens
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John_doe888_88
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  • Hi John. What have you tried, what are your thoughts about it? At the moment your post shows no evidence of effort. You can click "Edit" above, to improve your post. You should review the following, since you are new here: https://math.meta.stackexchange.com/q/9959 – 311411 May 14 '21 at 15:39

1 Answers1

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$|f(t+h)-f(t)|=|\lambda(E\cap[0,t+h])-\lambda(E\cap[0,t+h])|=\lambda(E\cap[t,t+h])\leq\lambda([t,t+h])=h$

Ciarán Ó Raghaillaigh
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