I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$
$$\int_*f:=\left\{\int h ; h \le f \text{ and h lower-continuous }\right\}$$
So a Lebesgue integral of a function $ f : \mathbb{R}^n \rightarrow \mathbb{R}$ exists $\int f \Leftrightarrow \int^*f = \int_*f$.
I am also allowed to use the following theorems:
- $L^p$ is a Banach space;
- Dominated convergence theorem;
- Monotone convergence theorem;
- $C^{\infty}$ is dense in the Lebesgue-functions.
But the huge problem is: We don't know what Borel-sets are and we don't have anything like measures so far. Therefore, all standard proofs of this theorem are not applicable to this situation. Hence, I wanted to find out whether anybody here knows a way how to do it?
Maybe I should say more about how this integral is defined:
Every semincontinuous function is the limit of a monotone sequence of continuous functions with finite support $g_n$. The integral over these kind of functions is defined via n-times 1 dimensional integration over all variables and then $\int h:= \lim_{n \rightarrow \infty} \int g_n$. (But this is probably not that relevant to this proof).
If anything is unclear, please let me know