How do we apply the dominated convergence theorem to conclude the proposed claim?

Question

Definition
Let ${f_{lambda}:lambdainLambda}$ be a collection of functions in $L^{1}(Omega,mathcal{F},mu)$. Then for each $lambdainLambda$, by the dominated convergence theorem and the integrability of $f_{lambda}$,
begin{align*}
a_{lambda}(t) := int_{{|f_{lambda}| > t}}|f_{lambda}|dmu rightarrow 0,,text{as},,trightarrow infty
end{align*}
The collection of functions ${f_{lambda}:lambdainLambda}$ in $L^{1}(Omega,mathcal{F},mu)$ is uniformly integrable if
begin{align*}
sup_{lambdainLambda}a_{lambda}(t) to 0,,text{as},,ttoinfty
end{align*}
My difficulties
I am mainly concerned about the preliminary part of the definition. To be more precise, how do we apply the dominated convergence theorem to conclude the proposed claim?
Once someone has helped me to understand it, I would great appreciate if someone could explain me the intuition of such definition.
Any help is appreacited.

Martin Argerami · Answer

Here $lambda$ is fixed, so it is irrelevant. We are also concerned only with $|f|$, so we may assume $fgeq0$. The situation then is that you have $fin L^1$, $fgeq0$. Let  $f_t=f,1_{{fgeq t}}$. Then $f_tto0$ pointwise. As $f_tleq f$, Dominated convergence applies and we have that
$$
lim_{ttoinfty}int_Omega f_t=int_{Omega}lim_{ttoinfty}f_t=0.
$$
"Uniformly Integrable" means that you can use the same $t$, for all $lambda$, to get the integrals to be as small as you want. Or, equivalently, that you can approximate $int_Omega |f_lambda|$ with $int_{{|f_lambda|leq t}}|f_lambda|$ with the same $t$ for all $lambda$.

d.k.o. · Answer

Note that $|f_{lambda}|1{{|f_{lambda}| > t}}le |f_{lambda}|in L^1$, and
$$
int_{{|f_{lambda}| > t}}|f_{lambda}|dmu=int |f_{lambda}|1{{|f_{lambda}| > t}}dmu.
$$

How do we apply the dominated convergence theorem to conclude the proposed claim?

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