Let $f$ be a positive measurable function.Show that $g $ defined by $g (x):= f (x)^{f (x)}$ is measurable.
I have tried my level best though I fail to grasp the problem in my own.Would anybody please help me by giving me some hint that leads to some satisfactory solution.Then that will really help me a lot.
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Consider $x\rightarrow\ln f(x)^{f(x)}=f(x)\ln f(x)$. Since $f$ is measurable and $\ln$ is continuous, then $\ln f$ is measurable, and so is the multiple $f\ln f$. And then $x\rightarrow\exp(f(x)\ln f(x))$ is measurable by the composition again by the continuous map $\exp$.