Tao's Introduction to Measure Theory asks me to prove the following result, named "Defect version of Fatou's lemma"
Let $(X,\mathcal{B},\mu)$ be a measure space, and let $f_1,f_2,\ldots : X \to [0,+\infty]$ be a sequence of unsigned absolutely integrable functions that converges pointwise to an absolutely integrable limit $f$. Show that $$ \int_X f_n\,d\mu - \int_X f\,d\mu - \|f - f_n\|_{L^1(\mu)} \to 0 $$ as $n \to \infty$.
My question is about a remark made afterwards:
Informally, this result tells us that the gap between the left and right-hand sides of Fatou's lemma can be measured by the quantity $\|f - f_n\|_{L^1(\mu)}$.
In what sense is the gap being "measured" by this sequence? Here's my thinking so far: Fatou's lemma says that $\int_X f\,d\mu \leq \liminf_{n\to\infty}\int_X f_n\,d\mu$, and we can deduce from the "defect Fatou's lemma" and superadditivity of lim inf that $$ \int_X f\,d\mu \geq \liminf_{n\to\infty}\int_X f_n\,d\mu - \limsup_{n\to\infty}\|f-f_n\|_{L^1}, $$ so therefore the "gap" in Fatou's lemma has the upper bound $$ \liminf_{n\to\infty}\int_X f_n\,d\mu - \int_X f\,d\mu \leq \limsup_{n\to\infty}\|f-f_n\|_{L^1}. $$ Is stating this upper bound what's meant by "measuring" the gap? Or can we say something more?