Research proof index

]]σ,τ]] = ⋃ᵢ (i, i+1] × {σ ≤ i < τ} as subsets of ℕ × Ω , a purely arithmetic identity on ℕ∞, valid for any σ, τ.

]]σ,τ]] is an elementary predictable set for bounded stopping times on ℕ (blueprint lem:elementaryPredictableSet_stochasticInterval). With τ ≤ n, ]]σ,τ]] = ⋃_{i < n} (i, i+1] × {σ ≤ i < τ} is a finite disjoint union of predictable rectangles, which is exactly the data of an ElementaryPredictableSet.

Along a strictly decreasing sequence u → x from the right, the regularized values r (u k) tend to the right limit of h at x along T' ⊇ T.

Along a strictly increasing sequence u → x from the left, the regularized values r (u k) tend to the left limit of h at x along T' ⊇ T.

The right-limit regularization inherits left limits of h along T.

A fibring identity in the $\tau$-minimizer setup. Let $X_1',X_2'$ be independent copies of $X_1,X_2$ and put $k=d[X_1;X_2]$. Then $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$.