# Question 14165: Block-Constant Values Give Conditioning Measurability [Return to catalogue page 71](https://ocme.evemisslab.com/en/questions/page/71/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r04-conditional-expectation-foundations/#q-20261014-advance-20261029-r04-conditional-expectation-0040) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0027-14165 · q-20261014-advance-20261029-r04-conditional-expectation-0040 ## 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](https://ocme.evemisslab.com/en/mko/mko-conditional-expectation/) ## Step-by-step derivation S001 · calculationBefore conditional model whole mass sum := sum(8/49, 6/49, 20/49, 15/49) After conditional model whole mass sum = 1 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore conditional model:raw aligned masses=[[8, 49], [6, 49], [20, 49], [15, 49]] After normalized masses=[8/49, 6/49, 20/49, 15/49];nonnegative=Yes;total=1 Conditions: none First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another model. Step identifiers and dependenciesRule: whole_finite_probability_law_not_uniform_assumption · Uses: G001, S001 S003 · impliesBefore same outcome indices=[0, 1, 2, 3];X values=[2, -3, 16, -2];block labels=[0, 1, 1, 2] After distinct conditioning labels in increasing order=[0, 1, 2] Conditions: none Each label fiber is a conditioning block.Labels need not be consecutive.Sort summaries by label but retain original outcome indices within blocks;labels are not array positions. Step identifiers and dependenciesRule: conditional_partition_same_space_original_indices · Uses: G001, S002 S004 · calculationBefore block0 mass := sum(8/49) After block0 mass = 8/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore block0 atom0 weighted X := mul(8/49, 2) After block0 atom0 weighted X = 16/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore block0 original weighted integral := sum(16/49) After block0 original weighted integral = 16/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore block0 positive-mass conditional mean := div(16/49, 8/49) After block0 positive-mass conditional mean = 2 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore block label=0;indices=[0];positive mass=8/49;weighted integral=16/49 After block-constant conditional value=2 Conditions: none Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block. Step identifiers and dependenciesRule: positive_block_mean_including_zero_mean · Uses: G001, S007 S009 · calculationBefore block1 mass := sum(6/49, 20/49) After block1 mass = 26/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore block1 atom1 weighted X := mul(6/49, -3) After block1 atom1 weighted X = -18/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore block1 atom2 weighted X := mul(20/49, 16) After block1 atom2 weighted X = 320/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore block1 original weighted integral := sum(-18/49, 320/49) After block1 original weighted integral = 302/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore block1 positive-mass conditional mean := div(302/49, 26/49) After block1 positive-mass conditional mean = 151/13 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore block label=1;indices=[1, 2];positive mass=26/49;weighted integral=302/49 After block-constant conditional value=151/13 Conditions: none Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block. Step identifiers and dependenciesRule: positive_block_mean_including_zero_mean · Uses: G001, S013 S015 · calculationBefore block2 mass := sum(15/49) After block2 mass = 15/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore block2 atom3 weighted X := mul(15/49, -2) After block2 atom3 weighted X = -30/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore block2 original weighted integral := sum(-30/49) After block2 original weighted integral = -30/49 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore block2 positive-mass conditional mean := div(-30/49, 15/49) After block2 positive-mass conditional mean = -2 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore block label=2;indices=[3];positive mass=15/49;weighted integral=-30/49 After block-constant conditional value=-2 Conditions: none Divide the weighted integral by mass only when mass is positive.A zero mean remains the determined numeric value0,not a null-block marker.Copy the same value to every atom in the block. Step identifiers and dependenciesRule: positive_block_mean_including_zero_mean · Uses: G001, S018 S020 · impliesBefore block=0;indices=[0];conditional values=[2] After distinct values in encounter order=[2];constant=Yes Conditions: none The conditioning sigma-algebra consists of block unions.For a block-constant function,every value-event preimage is a block union.A common null-block marker does not assert a unique version. Step identifiers and dependenciesRule: conditioning_measurability_every_complete_block · Uses: G001, S019 S021 · impliesBefore block=1;indices=[1, 2];conditional values=[151/13, 151/13] After distinct values in encounter order=[151/13];constant=Yes Conditions: none The conditioning sigma-algebra consists of block unions.For a block-constant function,every value-event preimage is a block union.A common null-block marker does not assert a unique version. Step identifiers and dependenciesRule: conditioning_measurability_every_complete_block · Uses: G001, S020 S022 · impliesBefore block=2;indices=[3];conditional values=[-2] After distinct values in encounter order=[-2];constant=Yes Conditions: none The conditioning sigma-algebra consists of block unions.For a block-constant function,every value-event preimage is a block union.A common null-block marker does not assert a unique version. Step identifiers and dependenciesRule: conditioning_measurability_every_complete_block · Uses: G001, S021 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 14740d082d297d06930bc7b1bc91f794b828a1de66ccd4be85ab23a0771fdd34 - candidate record SHA-256: 9fc72e796fbfe86dd5095316a0caaf95195a23701014d236800a98156e7a53f0 - semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe - question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r04-conditional-expectation-0040.json)