Simple function formula

Question

Let $f$ be a simple function in $(\mathbb{R},B(\mathbb{R}),\mu)$.

Can we write $f=\sum_{k=0}^n\alpha_k.1_{[a_k,b_k]}$ instead of $f=\sum_{k=0}^n\alpha_k.1_{A_k}$ with $A_k\in B(\mathbb{R})$?

Answer 1

No. Let $A_0=\mathbb Z$. Then $f=1_{A_0}$ has no representation of the form $f=\sum_{k=0}^n c_k 1_{[a_k,b_k]}$.