I have the function $g_n(x): n \sin (n^2x)$ for $x \in [0,1]$, $n \leq 2$ and I would like to calculate the essential supremum, i.e. $||g_n||_\infty= $inf$\{C \geq 0 : m(|g_n(x)| > C)=0\}$ where $m$ is a Lebesgue measure on $[0,1]$. I think that since $n \sin(n^2x) \leq n$ then $||g_n||_\infty=n$, but I'm not sure if I'm making some mistakes. Can somebody help me?
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3$\begingroup$ Isn't $n\sin(n^2x)$ continuous, so the essential supremum equals the maximum? $\endgroup$– SmileyCraftDec 13, 2018 at 14:13
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