For $1\le p<q<\infty$, show that neither $L^p(\mathbb{R^n})\subset L^q(\mathbb{R^n}) $ nor $L^q(\mathbb{R^n})\subset L^p(\mathbb{R^n})$.
Find a measurable function that belongs to $L^p(\mathbb{R^n})$ for all $p>1$ but not for $p=1$.
For Q1, I know some theorems shows the statement is true, since it contains measurable sets with arbitrary large and small measure, but I hope I can show it using an example. I also know how to construct a counterexample in $\mathbb{R}$. But I think this question asks me to find an example for all $n\in\mathbb{N}$? Is my understanding correct? If I only show an example in $\mathbb{R}$, is it OK?
The prototypical example in $\mathbb{R}$ is $1/|x|^\alpha$. Varying $\alpha$ puts this function in various $L^p(\mathbb{R})$-s. You can adapt this example of $\mathbb{R}^n$; you should integrate in polar coordinates to see which $\alpha$-s are needed.