I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue integration. I've seen short and slick proofs of the LDCT and MCT from scratch (in Bogachev's Measure Theory and Bass' Graduate Real Analysis/Rudin's Real and Complex Analysis, respectively). However, I was wondering if such a proof exists for Fatou's lemma. I've seen a couple of proofs that rely on neither the MCT nor the LDCT (in particular, in Royden and Fitzpatrick's Real Analysis, and on the Wikipedia page for Fatou's lemma), and while these proofs aren't too tricky or difficult to understand, they seem considerably longer than the proofs of the other two theorems.
So to summarize: Can someone supply me with a short and slick proof, perhaps even an outline of a proof for me to work through, of Fatou's lemma that does not rely on LDCT or MCT?
Thank you!
Best Answer
Let $\lambda<1$. For any $k$, denote $$B_k := \Big\{ x \in X ~\Big|~ \forall l \geq k,~ \lambda \liminf_n f_n(x) \leq f_l(x) \Big\}.$$ Then $$ \lambda \int_{B_k} \liminf_n f_n \leq \int_{B_k} f_k \leq \int_X f_k.$$ Take $\liminf_k$ in both side : $$ \lambda \liminf_k \left( \int_{B_k} \liminf_n f_n \right) \leq \liminf_k \int_X f_k.$$
The LHS is equal to $\lambda \int_X \liminf_n f_n$ because:
$(B_k)_k$ is increasing.
$\bigcup_k B_k=X$. Indeed for any $x \in X$, either $\liminf_n f_n(x)>0$ in with case $\lambda \liminf_n f_n(x) < f_k(x)$ for $k \gg 1$, or $\liminf_n f_n(x)=0$ in which case it is trivial.
Some property about measure will tell you that $\lim_k \int_{B_k} g = \int_X g$.