Define the characteristic function 𝜒𝐸 of a subset 𝐸 in ℝ by 

\(\rm \chi_E(x)=\left\{\begin{matrix}1,&if\ x\in E\\\ 0, &if\ x\notin E\end{matrix}\right.\)

For 1 ≤ 𝑝 < 2, let

𝐿^𝑝 [0, 1] = {𝑓:[0, 1] → ℝ ∶ 𝑓 is Lebesgue measurable and \(\rm \int_0^1|f(x)|^pdx<\infty\}.\)

Let 𝑓:[0, 1] → ℝ be defined by 

\(\rm f(x)=\Sigma_{n=1}^\infty \frac{2^n}{n^3}\chi\left[\frac{1}{2^{n+1},}\frac{1}{2^n}\right](x).\)

Consider the following two statements:

𝑃: 𝑓 ∈ 𝐿^𝑝 [0, 1] for every 𝑝 ∈ (1, 2).

𝑄: 𝑓 ∈ 𝐿¹ [0, 1].

Then