Any function in $L^p$ space is a linear combination of simple functions? True OR not?

Question

Any function in $L^p$ space is a linear combination of simple functions for $1<p<\infty$.

Is this true?

So any function in $L^p$ is measurable.

So any measurable function can be represented as a linear combination of simple functions.

So I think the statement should be true.

Is my understanding correct?

Answer 1

No, your understanding is not correct. All simple functions are measurable, but in most $\sigma$-algebras of interest, far more functions are also measurable. For instance, all continuous functions are measurable for the Lebesgue $\sigma$-algebra on $\mathbb{R}$, but only constant continuous functions are simple.

What is true is that all measurable functions can be realized as a.e. limits of sequences of simple functions.