Every simple functions in $L_p(\mathbb{T}) $ is a trigonometric polynomial.

Question

I know for a finite measure space $(\Omega, \mathcal{A}, \mu)$, Simple functions are dense in $L_p(\mu)$. Also it is true that, if we consider normalized Lebesgue measure on the unit circle then trigonometric polynomials are dense in $L_p(\mathbb{T})$.

I am new in this area and I cannot relate both the situation. So my question is the following :

Is it true that every simple functions in $L_p(\mathbb{T}) $ is a trigonometric polynomial?

Answer 1

No, of course not. Trigonometric polynomials are continuous. Non-constant simple functions are not. So no non-constant simple function is a trigonometric polynomial.