Why is quantile function of uniformly distributed random variable a random variable?

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 $$\mathbb{R}$$..