In my calculus lecture we introduced the following definitions. Let $(X, \mathcal{M}, \mu)$ be a measure space and $E$ a normed $\mathbb{R}-$Vektorspace.
- Let $A \in \mathcal{M}, \mu(A) < \infty$ and a finite partition $A = \biguplus_{i=1}^nA_i$ with $A_i \in \mathcal{M}$. $f: X \rightarrow E$ is called step-function for the partition $(A_i)$ of $A$ if $f|_{A_i}$ is constant for all $i = 1, \dots, n$ and $f(x) = 0$ for $x \not\in A$.
- $T(X,E) := \{f: X \rightarrow E\,|\, f\text{ is step-function for some partition } (A_i) \text{ of some }A \in \mathcal{M}\}$
- $||f||_1 := \int_X f\,d\mu$ defines a Seminorm on $T(X,E)$.
- $L^1(X, E) := \overline{T(X,E) / \{f \in T(X, E) : ||f||_1 = 0\}}$ - the formal completion of the quotient space
- $||\cdot||_1$ is a norm on $L^1(X,E)$.
- $\mathcal{L}^1(X, E)$ is now defined as the set of all $f: X \rightarrow E$ such that there exists an $L^1$-Cauchysequence $(f_n)_{n \in \mathbb{N}}$ in $T(X,E)$ with $f_n \rightarrow f$ (point-wise) everywhere except on a $\mu$-Nullset.
I think with $L^1$-Cauchysequence my professor means Cauchy-Sequence in $||\cdot||_1$-Norm. Now I don't quite understand how $\mathcal{L}^1(X,E)$ and $L^1(X, E)$ differ. The definition with the Cauchy-Sequence seems to produce something like the formal completion and everywhere except on a $\mu$-Nullset feels like the Quotient set, but I'm pretty sure $L^1$ and $\mathcal{L}^1$ are not the same.
How it this logically structured? $L^1$ was introduced before $\mathcal{L}^1$...