Manuscript VII-E@1 · VII-E.tex · proposition · b713cf3c

Proposition: Presentation-defect identity

A dictionary for distributional observations › Presentation defects before regulator removal · lines 1755–1792 · proof included or linked

VII-E@1#prop:presentation, VII-E@1#eq:finite-source, VII-E@1#eq:finite-target, VII-E@1#eq:presentation-defect

Suppose the two complete finite parts exist:

u_{fin} u_{fin}^{(P)} = ε ↓ 0 lim [u_{ε} - ℓ_{μ} (ε) r], = ε ↓ 0 lim [u_{ε}^{(P)} - ℓ_{μ} (ε) r^{(P)}] .

Then the presentation defect is

A_{P} := u_{fin}^{(P)} - P u_{fin} = ε ↓ 0 lim [u_{ε}^{(P)} - P u_{ε} - ℓ_{μ} (ε) (r^{(P)} - P r)] .

Proof. Continuity of $P$ gives

P [u_{ε} - ℓ_{μ} (ε) r] ⟶ P u_{fin} .

Subtract this convergence from eq:finite-target and collect terms.

Displayed formulas in this passage

VII-E@1#eq:finite-source · line 1758
$u_{fin} u_{fin}^{(P)} = ε ↓ 0 lim [u_{ε} - ℓ_{μ} (ε) r], = ε ↓ 0 lim [u_{ε}^{(P)} - ℓ_{μ} (ε) r^{(P)}] .$
VII-E@1#eq:finite-target · line 1758
$u_{fin} u_{fin}^{(P)} = ε ↓ 0 lim [u_{ε} - ℓ_{μ} (ε) r], = ε ↓ 0 lim [u_{ε}^{(P)} - ℓ_{μ} (ε) r^{(P)}] .$
VII-E@1#eq:presentation-defect · line 1770
$A_{P} := u_{fin}^{(P)} - P u_{fin} = ε ↓ 0 lim [u_{ε}^{(P)} - P u_{ε} - ℓ_{μ} (ε) (r^{(P)} - P r)] .$
unlabeled · line 1786
$P [u_{ε} - ℓ_{μ} (ε) r] ⟶ P u_{fin} .$