Question 14158: Block-Constant Values Give Conditioning Measurability

Original question and givens

Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 8/49, 15/49, 6/49], values X=[-5, -3, 9, 5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].

• G001 — Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 8/49, 15/49, 6/49], values X=[-5, -3, 9, 5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-conditional-expectation

Step-by-step derivation

conditional model whole mass sum := sum(20/49, 8/49, 15/49, 6/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[20, 49], [8, 49], [15, 49], [6, 49]]

normalized masses=[20/49, 8/49, 15/49, 6/49];nonnegative=Yes;total=1

same outcome indices=[0, 1, 2, 3];X values=[-5, -3, 9, 5];block labels=[0, 0, 1, 1]

distinct conditioning labels in increasing order=[0, 1]

block0 mass := sum(20/49, 8/49)

block0 mass = 4/7

block0 atom0 weighted X := mul(20/49, -5)

block0 atom0 weighted X = -100/49

block0 atom1 weighted X := mul(8/49, -3)

block0 atom1 weighted X = -24/49

block0 original weighted integral := sum(-100/49, -24/49)

block0 original weighted integral = -124/49

block0 positive-mass conditional mean := div(-124/49, 4/7)

block0 positive-mass conditional mean = -31/7

block label=0;indices=[0, 1];positive mass=4/7;weighted integral=-124/49

block-constant conditional value=-31/7

block1 mass := sum(15/49, 6/49)

block1 mass = 3/7

block1 atom2 weighted X := mul(15/49, 9)

block1 atom2 weighted X = 135/49

block1 atom3 weighted X := mul(6/49, 5)

block1 atom3 weighted X = 30/49

block1 original weighted integral := sum(135/49, 30/49)

block1 original weighted integral = 165/49

block1 positive-mass conditional mean := div(165/49, 3/7)

block1 positive-mass conditional mean = 55/7

block label=1;indices=[2, 3];positive mass=3/7;weighted integral=165/49

block-constant conditional value=55/7

block=0;indices=[0, 1];conditional values=[-31/7, -31/7]

distinct values in encounter order=[-31/7];constant=Yes

block=1;indices=[2, 3];conditional values=[55/7, 55/7]

distinct values in encounter order=[55/7];constant=Yes

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute block1 original weighted integral with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute block1 atom3 weighted X with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute block1 atom2 weighted X with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute block1 mass with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute block0 original weighted integral with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute block0 atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute block0 atom0 weighted X with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute block0 mass with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute conditional model whole mass sum with exact rational arithmetic, not rounded decimals.

Result

([-31/7, -31/7, 55/7, 55/7], [[0, [0, 1], [-31/7], Yes], [1, [2, 3], [55/7], Yes]], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.