Manuscript En01-A@1 · En01-A.tex · corollary · 558b972b

Corollary: Discrepancy decomposition

Operator-valued Möbius hierarchy and three orders · lines 1637–1656 · proof included or linked

En01-A@1#cor:decomposition

For every $J$ ,

X_{J} = - \emptyset \neq = K \subseteq J \sum Z_{K}, δ_{J} (Λ) \leq \emptyset \neq = K \subseteq J \sum ω_{K} (Λ) .

For a singleton,

X_{{j}} = - Z_{{j}}, δ_{{j}} (Λ) = ω_{{j}} (Λ) .

Proof. Subtract the inversion formula from $Z_{\emptyset} = ρ$ , then apply the triangle inequality.

Displayed formulas in this passage

unlabeled · line 1639
$X_{J} = - \emptyset \neq = K \subseteq J \sum Z_{K}, δ_{J} (Λ) \leq \emptyset \neq = K \subseteq J \sum ω_{K} (Λ) .$
unlabeled · line 1646
$X_{{j}} = - Z_{{j}}, δ_{{j}} (Λ) = ω_{{j}} (Λ) .$