Given $(\Omega, \mathcal F, \mu)$ measure space (In principle, $\mu$ is not necessarily a finite measure.). We know that
$$\int |f| d\mu < \infty \iff \lim_{b \to \infty} \int_{[\,|f|>b\,]} |f| d\mu =0$$ For the $\implies$, note that $|f| \mathbf{1}_{[\, |f| >b\,]}\leq |f|$. Since $|f|$ is integrable and $\lim_{b \to \infty} |f| \mathbf{1}_{[\, |f| >b\,]}=0\,\,$ a.s., we conclude the implication by the Dominated Convergence Theorem. For the converse, note that $|f| = |f| \mathbf{1}_{ [\, |f| > b \,]} + |f| \mathbf{1}_{ [\, |f| \leq b \,]} \leq |f| \mathbf{1}_{ [\, |f| > b \,]} + b$. Passing the integral, we can then choose $b$ large enough so that $\int |f| \mathbf{1}_{ [\, |f| > b \,]} d \mu < 1$ and conclude that $\int |f| d\mu \leq 1 + b \mu(\Omega)$. This show the integrability of $f$ under the hypothesis that $\mu(\Omega)< \infty$.
Now, we say that a squence of $(f_n)_{n \geq 1}$ is Uniform Integrable (UI) if $$\lim_{b \to \infty} \sup_n \int_{[\,|f_n|>b\,]} |f_n| d\mu =0$$
I would like to understand better why the definition of UI is not interesting for the case of $\mu(\Omega)=\infty$. The book Probability and Measure by Billingsley claims this, however, I quickly looked at the book An Introduction to Measure Theory, by Terence Tao, and it gives a notion of UI for measures that are not necessarily finite. I'm not really interested in knowing if both are equivalent, I'm just wanting to better understand Billingsley's comment saying that the case $\mu(\Omega)=\infty$ is not interesting;.