The function $\phi\colon t \mapsto \int_{0}^{1}f(x,t)\, dx$ is Borel Suppose $f\colon[0,1]\times [0,1] \rightarrow \mathbb{R} $. Functions $x \mapsto f(x,t)$ are integrable, functions $t \mapsto f(x,t)$ are continuous. I need to prove that the following function is Borel: $$\phi\colon t \mapsto \int_{0}^{1}f(x,t)\,dx.$$
I tried to use monotone class theorem for $E=\{B \in \mathcal{B}(\mathbb{R})\mid \phi^{-1}(B) \in \mathcal{B}([0,1]) \} \subset \mathcal{B}(\mathbb{R})$. I need to show that $E=\mathcal{B}(\mathbb{R})$. The only thing I managed to prove is that $E$ is an algebra. So, the idea seems wrong.
Please give me any ideas how to solve the problem. Thank you for any hints!

The condition is so loose that it had me confused in many ways( it also made me question my understanding of measurability )
Your desired measurability ( of integrals) is already covered in (and also plays a part of ) the demonstration of Fubini's theorem. 
Moreover, as you know, the continuity property is still too much for only proving measurability.

If you want to reproduce that theorem, you have realized the right way to it, that is, to use the monotone class theorem( but in a more general setting). Let 
$$S:=\{ A \in \mathcal{B}([0,1])\otimes \mathcal{B}([0,1]) | f:= \mathbb{1}_{A} \text{ sasisties the respective }\phi \text{ is measurable}\}$$
then, this family $S$ is a Dynkin system, containing all product $1_{B\times C}$(where $B,C \in \mathcal{B}([0,1])$).
From then you can prove your desired results with integrable positive functions.
Etc.


Discussion

*

*given your constraint, if we suppose further that $f$ is bounded (1), you can even prove the continuity of $\phi$ by using dominated convergence theorem.

*The boundedness (1) perhaps can be relaxed, but more elaborate care should be put in.

