Measurable function in the context of Open set

In mathematics, the layer cake representation of a non-negative, real-valued measurable function ${\displaystyle f}$ defined on a measure space ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ is the formula

for all ${\displaystyle x\in \Omega }$, where ${\displaystyle 1_{E}}$ denotes the indicator function of a subset ${\displaystyle E\subseteq \Omega }$ and ${\displaystyle L(f,t)}$ denotes the (strict) super-level set:

In mathematics, a square-integrable function, also called a quadratically integrable function or ${\displaystyle L^{2}}$ function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line ${\displaystyle (-\infty ,+\infty )}$ is defined as follows.

One may also speak of quadratic integrability over bounded intervals such as ${\displaystyle [a,b]}$ for ${\displaystyle a\leq b}$.