Lebesgue integrability via a partition of the range

Let $\mu$ denote the Lebesgue measure on $\mathbb R$. Suppose that $f$ is a non-negative Lebesgue measurable function on $\mathbb R$. Let $0=a_0<a_1<a_2<\cdots$ be an unbounded sequence such that $a_{n+1}\le ca_n$ for some real number $c$ and for all $n\ge1$. Let

$$A_k=\{x\in\mathbb R\mid a_k\le f(x)<a_{k+1}\}$$

for each $k\ge0$. Which of the following statements are true?