Notation of this set in a set? I am currently struggeling with the following notation: 
For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$:
$M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ||f||_p\} \ge \epsilon\}$.
I do not know how to interpret this set $M(p,\epsilon)$ cause this inner set confuses me. Does anybody here know how to interpret this?
 A: It is the set of all $p$-integrable functions with the property that the set of points where the absolute value of the function exceeds the $\epsilon$-fold $p$-norm of the function has measure at least $\epsilon$.
A: $\;M\;$ is the set of all the functions $\;f\;$  in $\;L^p\;$ s.t. that the measure of of the set $\;\{x\;;\;|f(x)|\ge\epsilon\cdot||f||_p\}\;$ is greater than or equal to $\;\epsilon\;$ . 
Write it with colors if it makes things easier:
$$M(p,\epsilon):=\left\{f\in L^p\;:\;m\color{red}\{|f(x)|\ge\epsilon\cdot||f||_p\color{red}\}\ge\epsilon\right\}$$
A: Remember: technically, the expression "$L^p$" is ambiguous - really, for any given measure space $(X,\Sigma,m)$, we define the collection of functions $L^p(X,\Sigma,m)$, namely
$$L^p(X,\Sigma,m)=\left\{\text{measureable functions }f:X\to\mathbb{R}\;\middle\vert\;\left(\int_X|f|^p\,dm\right)^{1/p}<\infty\right\}\bigg/\text{a.e.}$$
So the "$m$" in the expression you're asking about,
$$M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ||f||_p\} \ge \epsilon\}$$ is the measure from the (implicit) measure space. We are taking the measure of the set
$$\{x:|f(x)|\geq\epsilon\|f\|_p\}$$
where "$x$" is (again implicitly) intended to range over the space $X$. And clearly, the interpretation of 
$$\{x\in X:|f(x)|\geq\epsilon\|f\|_p\}$$
is

the subset of $X$ consisting of those points $x\in X$ for which $|f(x)|$ is greater than or equal to the quantity $\epsilon\|f\|_p$.

So in the expression you're asking about,
$$\begin{align*}
M(p,\epsilon)&=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ||f||_p\} \ge \epsilon\}\\\\
&=\{f \in L^p(X,\Sigma,m):m\left(\{x\in X:|f(x)| \ge \epsilon ||f||_p\}\right) \ge \epsilon\}
\end{align*}$$
we're given a measure space $(X,\Sigma,m)$, an $\epsilon \in (0,1)$, and a $p \in (0,\infty)$, and we're taking the collection of $f\in L^p(X,\Sigma,m)$ such that the set
$$\{x\in X:|f(x)| \ge \epsilon ||f||_p\}$$
has measure $\ge\epsilon$.
