On Page 11 in slides, the process $$ \sqrt{n}(\phi(F_n) - \phi(F)) = \phi_F'(\mathbb{G}_{n,F}) + o_P(1). $$ We know that $\mathbb{G}_{n,F}$ converges weakly to is the F-Brownian bridge, defined as $\mathbb{G}_{F} = \mathbb{G}_{\lambda} \circ F $. We also have that $\phi(F) = F^{-1}(p)$ is the quantile function for $p\in[0,1]$.
I need to calculate the expectation of the above equation $$ \sqrt{n} P (\phi(F_n) - \phi(F)) = \int \phi_F'(\mathbb{G}_{n,F}) {\rm d}F. $$ I start by noting that by continuous mapping theorem $$ \int \phi_F'(\mathbb{G}_{n,F}) {\rm d}F \to \int \phi_F'(\mathbb{G}_{F}) {\rm d}F $$ in probability. By definition of Brownian bridge I have that $\mathbb{G}_{F}$ is zero-mean Gaussian process with a finite variance. I also know that $\phi_F'$ is continuous on the domain. Can I then claim that $$ \int \phi_F'(\mathbb{G}_{F}) {\rm d}F = \phi(\mathbb{G}_{F}) $$ ? It seems easy but somehow I miss the interpretation of $\phi(\mathbb{G}_{F})$. What can I say about its distribution? Thanks for you help!