Question 14165: Block-Constant Values Give Conditioning Measurability

Original question and givens

Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 6/49, 20/49, 15/49], values X=[2, -3, 16, -2], and conditioning-block labels=[0, 1, 1, 2]. 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=[8/49, 6/49, 20/49, 15/49], values X=[2, -3, 16, -2], and conditioning-block labels=[0, 1, 1, 2]. 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(8/49, 6/49, 20/49, 15/49)

conditional model whole mass sum = 1

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

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

same outcome indices=[0, 1, 2, 3];X values=[2, -3, 16, -2];block labels=[0, 1, 1, 2]

distinct conditioning labels in increasing order=[0, 1, 2]

block0 mass := sum(8/49)

block0 mass = 8/49

block0 atom0 weighted X := mul(8/49, 2)

block0 atom0 weighted X = 16/49

block0 original weighted integral := sum(16/49)

block0 original weighted integral = 16/49

block0 positive-mass conditional mean := div(16/49, 8/49)

block0 positive-mass conditional mean = 2

block label=0;indices=[0];positive mass=8/49;weighted integral=16/49

block-constant conditional value=2

block1 mass := sum(6/49, 20/49)

block1 mass = 26/49

block1 atom1 weighted X := mul(6/49, -3)

block1 atom1 weighted X = -18/49

block1 atom2 weighted X := mul(20/49, 16)

block1 atom2 weighted X = 320/49

block1 original weighted integral := sum(-18/49, 320/49)

block1 original weighted integral = 302/49

block1 positive-mass conditional mean := div(302/49, 26/49)

block1 positive-mass conditional mean = 151/13

block label=1;indices=[1, 2];positive mass=26/49;weighted integral=302/49

block-constant conditional value=151/13

block2 mass := sum(15/49)

block2 mass = 15/49

block2 atom3 weighted X := mul(15/49, -2)

block2 atom3 weighted X = -30/49

block2 original weighted integral := sum(-30/49)

block2 original weighted integral = -30/49

block2 positive-mass conditional mean := div(-30/49, 15/49)

block2 positive-mass conditional mean = -2

block label=2;indices=[3];positive mass=15/49;weighted integral=-30/49

block-constant conditional value=-2

block=0;indices=[0];conditional values=[2]

distinct values in encounter order=[2];constant=Yes

block=1;indices=[1, 2];conditional values=[151/13, 151/13]

distinct values in encounter order=[151/13];constant=Yes

block=2;indices=[3];conditional values=[-2]

distinct values in encounter order=[-2];constant=Yes

Reverse checks

• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute block2 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute block2 original weighted integral with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute block2 atom3 weighted X with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute block2 mass with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute block1 original weighted integral 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 atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute block1 mass with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute block0 original weighted integral 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

([2, 151/13, 151/13, -2], [[0, [0], [2], Yes], [1, [1, 2], [151/13], Yes], [2, [3], [-2], 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.