Question 14365: A Measurable Event Is a Union of Whole Current Blocks

Original question and givens

Use the same finite carrier Fin(6)={0,…,5} at all times, with complete partitions=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 11, 10, 10]] in time order starting at 0. Equal labels in one row mean the same information block; numerical label size and labels in other rows carry no extra meaning. A filtration requires every later partition to refine its predecessor. Event measurability and process adaptedness are separate checks; no natural-filtration, independence or Markov conclusion is inferred. time=2, event=[1, 2]. Normalize the event as increasing distinct outcome indices. Give (time,event,measurable?,event_must_be_union_of_time_blocks).

• G001 — Use the same finite carrier Fin(6)={0,…,5} at all times, with complete partitions=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 11, 10, 10]] in time order starting at 0. Equal labels in one row mean the same information block; numerical label size and labels in other rows carry no extra meaning. A filtration requires every later partition to refine its predecessor. Event measurability and process adaptedness are separate checks; no natural-filtration, independence or Markov conclusion is inferred. time=2, event=[1, 2]. Normalize the event as increasing distinct outcome indices. Give (time,event,measurable?,event_must_be_union_of_time_blocks).

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-finite-filtration

Step-by-step derivation

same carrier Fin(6);complete partitions in original time order=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 11, 10, 10]]

equal labels mean same information block only within that row

time=1;later block label=0;indices=[0, 1, 2]

earlier block labels in encounter order=[0];contained in one earlier block=Yes

time=1;later block label=1;indices=[3, 4, 5]

earlier block labels in encounter order=[0];contained in one earlier block=Yes

time=2;later block label=0;indices=[0]

earlier block labels in encounter order=[0];contained in one earlier block=Yes

time=2;later block label=1;indices=[1, 2]

earlier block labels in encounter order=[0];contained in one earlier block=Yes

time=2;later block label=11;indices=[3]

earlier block labels in encounter order=[1];contained in one earlier block=Yes

time=2;later block label=10;indices=[4, 5]

earlier block labels in encounter order=[1];contained in one earlier block=Yes

all nesting failures in time,a,b order=[]

filtration=Yes;per-time measurability checks remain separate

time=2;block label=0;indices=[0];normalized event=[1, 2]

selected block indices=[];whole-or-empty=Yes

time=2;block label=1;indices=[1, 2];normalized event=[1, 2]

selected block indices=[1, 2];whole-or-empty=Yes

time=2;block label=11;indices=[3];normalized event=[1, 2]

selected block indices=[];whole-or-empty=Yes

time=2;block label=10;indices=[4, 5];normalized event=[1, 2]

selected block indices=[];whole-or-empty=Yes

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, [1, 2], Yes, event_must_be_union_of_time_blocks)

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.