Is there any sequence ${f_n}$ that converges to 0 in measure but $g_n=\frac{f_1 + \cdots f_n}{n}$ does not

Asked Dec 29 '24 at 13:38

Active Dec 29 '24 at 13:51

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Consider [0, 1] with the Lebesgue measure.

Is there any sequence ${f_n}$ of nonnegative integrable functions defined on [0, 1] such that $f_n$ converges to 0 in measure, but the sequence ${g_n}$ defined by $g_n =\frac{f_1 + \cdots + f_n}{n}$ does not converge to 0 in measure?

edited Dec 29 '24 at 13:51

asked Dec 29 '24 at 13:38

IceAgeV

1

Welcome to Math.SE! Please provide context to your question; in particular, why do you believe a counterexample may exist? Are you aware of the corresponding result for sequences in $\mathbb R$? – ktoi Dec 29 '24 at 13:45

Is there any sequence ${f_n}$ that converges to 0 in measure but $g_n=\frac{f_1 + \cdots f_n}{n}$ does not

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