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Show that $$ f\colon\mathbb{R}\to\mathbb{R}, x\mapsto\chi_{[1,\infty)}(x)\frac{1}{x}\mbox{ and }g\colon [1,\infty)\to\mathbb{R}, x\mapsto\frac{1}{x} $$ are measurable.

Concerning $f$, I guess $\mathcal{B}(\mathbb{R})\setminus\mathcal{B}(\mathbb{R})$ - measurability is meant?

If yes: For all $r\in\mathbb{R}$ it is $$ f^{-1}((-\infty,r])=\begin{cases}\mathbb{R}, & r\geq 1\\ \emptyset, & r<0\\ [1/r,\infty), & 0<r<1\\ [1,\infty)^C, & r=0\end{cases} $$ and this are all Borelsets, so $f$ is measurable.


Concerning $g$ it probably is meant $\mathcal{D}\setminus\mathcal{B}(\mathbb{R})$ - measurability with $\mathcal{D}=\left\{B\subseteq [1,\infty) : B\in\mathcal{B}(\mathbb{R})\right\}$?

If yes, it is $\forall r\in\mathbb{R}$: $$ g^{-1}((-\infty,r])=\begin{cases}[1,\infty), & r\geq 1\\ \emptyset, & r\leq 0\\ [1/r,\infty), & 0<r<1\end{cases} $$ and this are all sets that are in $\mathcal{D}$, so $g$ is measurable in the sense how I understood it.

Wish you a nice weekend and would be great to get a kind of feedback!

With kind regards

math12

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