$\mathbb{E}[f(T)]$ as an integral over $E$ and $\Omega$ with change of variables

I am working on the following exercise:

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let $(E, \mathcal{E})$ and $(F, \mathcal{F})$ be measurable spaces. Let$$ Y : (\Omega, \mathcal{A}, \mathbb{P}) \to (E, \mathcal{E}) $$be a random variable. Let$$ T : (E, \mathcal{E}, \mathbb{P}_y) \to (F, \mathcal{F}) $$be a measurable mapping, and let$$ f : (F, \mathcal{F}) \to ([0, \infty], \mathcal{B}(\mathbb{R})) $$be a measurable function.

Then $T \circ Y : (\Omega, \mathcal{A}, \mathbb{P}) \rightarrow (F, \mathcal{F})$

I have to express $\mathbb{E}[f(T)]$ as an integral over $E$ and $\Omega$.

2. An integral over $\Omega$, using $\mathbb{P}$, i.e.,$$ \mathbb{E}[f(T)] = \int_\Omega f(T(Y(\omega))) \, d\mathbb{P}(\omega), $$

I've tried to express it over E. I know $T$ and $T \circ Y$ have the same distribution and T's domain is E. Therefore, i believe it should be, using LOTUS:

$$\mathbb{E}[f(T)]=\int_E f(T(x))d \mathbb{P}_y$$

with $f(T(x))$ since the domain of $f$ is $E$. I'm having trouble and i don't think this is correct? I've tried to research change of variables with no luck.