how do you call this type of convergence?

Question

in $L^1(A)$ equipped with the norm $\| f\|_1 = \int_{A} |f| d\lambda$ where $\lambda$ is the Lebesgue measure.

let $(g_n)$ be a sequence of functions if there exists $g \in L^1(A)$ such that $\| g_n - g \|_1 \to 0$

then $g_n$ converges to $g$.

what's the usual name of this type of convergence ?

Answer 1

Since the question has been answered in the comments, I'm going to post it as community wiki.

It's called convergence in $L^1$ norm, and it is denoted as $\lVert g_n - g \rVert_1 \xrightarrow[n\to\infty]{} 0$ or $g_n \xrightarrow[n\to\infty]{L^1}g$.