Is this valid on $\mathbb R^n$ when $\mu$ is the Lebesgue measure? What about on arbitrary measure spaces?
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Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Jun 23 '23 at 19:39
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This has been asked and answered many times before. See https://math.stackexchange.com/questions/1357860/for-each-epsilon-0-there-is-a-delta-0-such-that-whenever-ma-delta?rq=1 for example. – FZan Jun 23 '23 at 20:27