What does $w(t) ∈ \mathcal{L}^1$ mean?

Question

I am reading a paper where it is mentioned the following:

The function $w(t) ∈ \mathcal{L}^1$, where $w(t) ≥ 0$

so what does $\mathcal{L}^1$ mean here?

Answer 1

In most contexts in math, the set $\mathcal{L}_1$ is the set of all such functions from a space $X$ to either $\mathbb{R}$ or $\mathbb{C}$ that are absolutely integrable:

$$ \mathcal{L}_1 = \left\{ f:X \to \mathbb{C} : \int_X |f| dx < \infty \right\} $$

In measure theory, the functions are measurable and the integral is $\int_X |f| d\mu$