Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$

Question

I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows.

Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a measure space $(X,\mu)$ such that $f_n \to f$ in $L^{q,s}$. That is, $\|f_n - f\|_{L^{q,s}}\to 0$ as $n \to \infty$. Then $$\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$$

The source I have says it follows from the fact that $|f| \leq \liminf\limits_{n\to\infty} |f_n|$ $\mu$-a.e. implies $f^{*} \leq \liminf\limits_{n\to\infty} f_n^{*}$, where $f^{*}$ is the decreasing rearrangement of $f$. However, I don't see how this helps since $f_n$ may not convergence pointwise to $f$. Also, my attempts at simply trying to compute it directly using the definition below have failed. Any help is greatly appreciated! Some background material that may help:

$$d_{f}(\alpha) = \mu\big(\{x \in X : |f(x) | \geq \alpha\}\big)$$ $$f^{*}(t) = \inf \{\alpha > 0 : d_f(\alpha) \leq t\}$$ $$\|f\|_{L^{q,s}} = \left(\int_{0}^\infty \left(t^{\frac{1}{q}} f^{*}(t) \right)^s \frac{dt}{t}\right)^{1/s}$$

Edit: See my answer below.

Answer 1

I have figured it out, so I thought I would provide the proof in case someone ever needs it.

Proof. From general theory (see for example Section 1.4 of Classical Fourier Analysis by L. Grafakos) we know that $$ \sup_{\alpha > 0} \alpha \big(d_{f^n-f}(\alpha)\big)^{1/s} = \sup_{t > 0} t^{1/s} (f_n - f)^{*}(t) \leq \left(\dfrac{s}{q}\right)^{1/s} \| f_n - f\|_{L^{q,s}}. $$ and by assumption this goes to zero. Therefore, $f_n \to f$ in measure and so there is a subsequence $\{f_{n_k}\}$ of $\{f_n\}$ so that $f_{n_k} \to f$ $\mu$-a.e. Now the result mentioned above can be applied to conclude that $$ f^{*} \leq \liminf_{n_k\to\infty} f_{n_k}^*. $$ Hence, this result and Fatou's lemma yields $$ \begin{align*} \| f \|_{L^{q,s}} &= \left(\int_0^\infty \left(t^{1/q} f^*(t)\right)^{s} \frac{dt}{t} \right)^{1/s}\\ &\leq \left(\int_0^\infty \left(t^{1/q} \liminf_{n_k\to\infty}f_{n_k}^*(t)\right)^{s} \frac{dt}{t} \right)^{1/s} \quad \mbox{(result above)}\\ &\leq \liminf_{n_k\to\infty}\left(\int_0^\infty \left(t^{1/q} f_{n_k}^*(t)\right)^{s} \frac{dt}{t} \right)^{1/s} \quad \mbox{(Fatou's Lemma)}\\ &= \lim_{n_k \to \infty} \|f_{n_k}\|_{L^{q,s}} \\ &= \lim_{n\to\infty} \|f_{n}\|_{L^{q,s}} \end{align*} $$ where the final equality follows from the fact that $L^{q,s}$ is a complete space.