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I came across a proof for the following proposition. However, there is one part of the proof which I don't understand. The proposition and proof are as follow.

Proposition: A nonempty family $\mathcal{X} ⊆ L^0$ is uniformly integrable if and only if there exists a test function of uniform integrability $ϕ$ such that $\sup_{ X∈\mathcal{X}} \mathbb{E}[ϕ(|X|)] < ∞$. Moreover, if it exists, the function ϕ can be chosen in the class of nondecreasing convex functions.

Proof (partial): Suppose, first, that this condition holds for some test function of uniform integrability and that the value of the supremum is $0 ≤ M < ∞$. For $n > 0$, there exists $C_n ∈ \mathbb{R}$ such that $ϕ(x) ≥ nMx$, for $x ≥ C_n$. Therefore, $$M ≥ \mathbb{E}[ϕ(|X|)] ≥ \mathbb{E}[ϕ(|X|)\textbf{1}_{{|X|≥C_n}}] ≥ nM\mathbb{E}[|X| \textbf{1}_{{|X|≥C_n}}],$$

for all $X ∈ \mathcal{X}$ . Hence,

$$\sup_{X∈\mathcal{X}} \mathbb{E}[|X| \textbf{1}_{{|X|≥C_n}}] ≤ 1/n$$ and the uniform integrability of X follows.

In the above proof, can someone explain to me the part "For $n > 0$, there exists $C_n ∈ \mathbb{R}$ such that $ϕ(x) ≥ nMx$, for $x ≥ C_n$."?

jason
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user372128
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  • By the way this is in reference to the proof of Proposition 12.6 from here https://www.ma.utexas.edu/users/gordanz/notes/uniform_integrability.pdf – jason Aug 22 '18 at 05:42
  • This follows immediately from the author's definition of "test function" – Tim kinsella Dec 10 '20 at 10:08

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