Let $f$ be an integrable function over $E$. Then given $ε > 0$, there is a simple function $φ$, step function $ψ$, and a continuous function $g$ supported on a finite measure set such that $\int_E|f − φ| < ε$, $\int_E|f − ψ| < ε$, $\int_E|f − g| < ε$.
Now I know one result that if Let $f$ be a measurable function over $E$. Then given $ε > 0$, there is a simple function $φ$, step function $ψ$, and a continuous function $g$ supported on a finite measure set such that $|f − φ| < ε$, $|f − ψ| < ε$, $|f − g| < ε$ a.e .
Now for $f$ integrable $\int_E f=\int_E f^+ - \int_E f^- <\infty$ where $f^+$ and $f^-$ are non-negative functions so from the definition of non-negative function integrable $\int_E f=sup_{0\leq h\leq f} \int_Eh$ where $h:E \to \Bbb R$ bounded, supported on a finite measurable set.
So my question is
- Is $h$ measurable here?( Because I have an intution that $h$ is measurable here)
- If it is then how to prove the main question?