Subsampling estimates of the Lasso distribution.

2.2 Convergence in distribution 7
and
E ∗ (T ) = sup {E(U)|U ≤ T, U : Ω → R measurable and E(U) exists} (2.2.0.5)
respectively.
(ii) The outer probability and the innner probability of an arbitrary set B ⊂ Ω are
defined as
and
respectively.
P ∗ (B) = inf {P (A)|A ⊃ B, A ∈ A} (2.2.0.6)
P ∗ (B) = sup {P (A)|A ⊂ B, A ∈ A} (2.2.0.7)
It turns out that outer/inner integrals and outer/inner probabilities are indeed attained
at measurable maps and sets, respectively, as stated in
Lemma 2.2.0.5. (Van der Vaart and Wellner, 1996, Lemma 1.2.1) Let (Ω, A, P ) be a
probability space. For an arbitrary map T : Ω → R, there exist measurable functions
T ∗ , T ∗ : Ω → R with
(i) T ∗ ≥ T ,
(ii) T ∗ ≤ U a.s. for every measurable U : Ω → R with U ≥ T a.s.,
(iii) T ∗ ≤ T ;
(iv) T ∗ ≥ U a.s. for every measurable U : Ω → R with U ≤ T a.s.
For every such T ∗ and T ∗ , it holds that E ∗ (T ) = E(T ∗ ) and E ∗ (T ) = E(T ∗ ), respectively,
provided that E(T ∗ ), respectively E(T ∗ ) exists.
We call T ∗ and T ∗ minimal measurable majorant and maximal measurable minorant
of T , respectively.
2.2.1 Weak convergence in metric spaces
In the remaining, let (D, d) nad (E, e) denote metric spaces.
Definition 2.2.1.1. (Weak convergence) Let (Ω n , A, P n ) , n ∈ N, be probability spaces
and let {X n : Ω n → D} n be a sequence of arbitrary random maps. {X n } n converges weakly
to a Borel measure L if
This is denoted by X n L.
∫
E ∗ (f(X n )) → fdL,
for every f ∈ C b (D).