Its an if and only if but I have one direction down. So suppose $(X,\mathbf{A},\mu)$ is a measure space and $X$ contains sets of arbitrarily small positive measure. Choose $A_1 \in \mathbf{A}$ such that $\mu(A_1) \in (0,1]$ And inductively for $n$ choose $A_n \in \mathbf{A}$ such that
$$\mu(A_n) \in (0,\frac{1}{3}\mu(A{n-1})], \mu(A_n) \in (0,\frac{1}{3^n}].$$
And more generally,
$$\mu(A_{n+k}) \in (0,\frac{1}{3^k}\mu(A_n)],$$
for each $n,k$. Then put $E_n=A_n \setminus _{k=1}^\infty A_{n+k}$, then if $a_n \geq 0$ and $f=\sum_n a_n \chi_{E_n}$, then
$$\int \vert f\vert^p = \sum_n a_n^p\mu(E_n).$$
Putting $a_n=\mu(E_n)^{-\frac{1}{q}}$, one has
\begin{align} \int \vert f \vert^p &=\sum_n (\mu(E_n)^{-\frac{1}{q}})^p\mu(E_n)\\ &\leq \sum_n \frac{1}{3^{n(1-\frac{p}{q})}\\ &<\infty. \end{align} Thus $f \in L^p$. But $\int \vert f \vert^q$ is $\sum_n 1 = \infty$ thus $f \not\in L^q$. Does this construction work? Also why is my begin align not working
