Real Analysis – Dominated Convergence Theorem

I have trouble understanding the proof in the paper Learning about the temporal evolution of spatial dependence
Generalized spatiotemporal Gaussian process models
,

Theorem 2.1 on page 33 uses the dominated convergence theorem (DCT) to show that the following series converges $ L ^ 1 ( mathcal {Z} times mathcal {Z}) $ feel where $ mathcal {Z} = mathbf {X} times mathcal {T} $
$$ sum_ {l = 1} ^ infty lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;) $$
by showing that
$$ sum_ {l = 1} ^ infty Bigg vert int _ { mathcal {Z}} int _ { mathcal {Z}} lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t)) lambda_l (t)) phi_l ( mathbf {x}) phi_l ( mathbf {x}) d mathbf {z} d mathbf {z} & # 39; Bigg vert < infty $$
Where $ mathbf {z} = ( mathbf {x}, t) $ and $ mathbf {z} = ( mathbf {x} & # 39 ;, t & # 39;) $,

For me, however, the direct application of DCT would be the limits of
$$ sum_ {l = 1} ^ infty int _ { mathcal {Z}} int _ { mathcal {Z}} Big vert lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t #) lambda_l (t #) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Big vert d mathbf {z} d mathbf {z} & # 39; < infty $$

with a monotonous sequence of dominating functions
$$ sum_ {l = 1} ^ L Big vert lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Big vert ge Bigg vert sum_ {l = 1} ^ L lambda_l (t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Bigg vert quad forall L in mathbf {N} $$

have the following condition
$$ int _ { mathcal {Z}} int _ { mathcal {Z}} sum_ {l = 1} ^ L Big vert lambda_l (t) mathcal {C} _ { mathcal { T}} (t, t)) lambda_l (t)) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39;
Big vert d mathbf {z} d mathbf {z} & # 39; $$

$$ xrightarrow () {L rightarrow infty} sum_ {l = 1} ^ infty int _ { mathcal {Z}} int _ { mathcal {Z}} large vert lambda_l ( t) mathcal {C} _ { mathcal {T}} (t, t & # 39;) lambda_l (t & # 39;) phi_l ( mathbf {x}) phi_l ( mathbf {x} & # 39)
Big vert d mathbf {z} d mathbf {z} & # 39; < infty $$

Do I miss something?

Thank you in advance.