I have the quantilfe function $F^{-1}$ of a random variable which is defined as:

$F^{-1}: ]0, 1[ \ni u \rightarrow F^{-1}(u) = inf\{x: F(x) \geq u\} \in \mathbb{R}$.

Now I can define a new random variable $\xi := F^{-1}(\eta)$, where $\eta$ is a uniformly distributed random variable.

I know the actual proof, but I am wondering how the random variable $\xi$ which is actually $F^{-1}(\eta)$ matches the defintion of a random variable:

$\zeta : (\Omega, A, \mathbb{P}) \ni\omega \rightarrow \zeta(\omega) \in (\mathbb{R}, \mathbb{B}^{*})$. (I am referring to real-valued random variables)

In my case it is clear that $\eta$ maps into $(\mathbb{R}, \mathbb{B}^{*})$. In order to get the composite function $F^{-1}(\eta), F^{-1}$ has to map from $(\mathbb{R}, \mathbb{B}^{*})\rightarrow(\mathbb{R}, \mathbb{B}^{*})$ which should then result in the final $A-\mathbb{B}^{*}-measurable$ function that is a random variable.
However, referring to the definition of $F^{-1}$, the pre-image of $F^{-1}$ is $]0,1[$ and the image is $\mathbb{R}$..


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