I am trying to show that $h_n(x) = n \log\left(1+\left(\frac{f}{n} \right)^{1/2} \right)$ are Lebesgue-measurable function given $f$ a positive Lebesgue-measurable function.
I have a definition that says that $f$ is Lebesgue measurable iff $\{f < \alpha\}$ is measurable for every $\alpha \in \mathbb{R}$. So $\{h_n < \alpha\} = \{1+(\frac{f}{n})^{1/2} < e^{\alpha/n}\} = \{(\frac{f}{n})^{1/2} < e^{\alpha /n} - 1\} = \{f < n(e^{\alpha/n} - 1)^2\}$. Which I suppose that is measurable, because $f$ is. Is this procedure correct or is there a more straightforward way to show what I want?
To put it simply, in this case the composition of measurable functions is measurable.
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