Use the standard Cantor distribution affinely carried onto [-1,2], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
G001 — Use the standard Cantor distribution affinely carried onto [-1,2], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).
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.
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ORIGINAL affine Cantor support[-1,2];positive width W=3
After
take independent fair digits B_k in{0,1},Y=sum_(k>=1)2*B_k/3^k and X=L+W*Y;the geometric tail converges and X lies in the affine Cantor support
Conditions: none
This defines the whole Cantor law,not the piecewise-linear constructor or a finite CDF sample.The convergent ternary0/2 series has probability1 on Cantor support,and the positive affine transformation is nondegenerate.
ORIGINAL support width W=3;level n ternary construction uses2^n intervals of length W/3^n
After
total level-n length=W*(2/3)^n tends0,so support has Lebesgue measure0;any singleton has probability<=2*2^(-n) for EVERY n,hence atom probability0;CDF jumps are0
Conditions: none
The nested level sets contain the support and their total length tends0,proving Lebesgue-null support.A fixed point is bounded by at most two relevant cylinder masses2^(-n).The bound for every n proves no atoms,not sampling finitely many points.
ORIGINAL affine law on[-1,2] has support N with Lebesgue measure0 and probability1;all singleton probabilities0
After
CDF_continuous=Yes;atomless=Yes;has_density=No;singular=Yes;any Lebesgue density would integrate to0 over N,contradicting probability1
Conditions: none
CDF jumps equal singleton masses;an everywhere atomless law has a continuous CDF.The full-probability null support proves singularity and rules out a Lebesgue density,since its integral over that support would be0.This is a particular counterexample to continuity/atomlessness implying density.
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 — Recompute Cantor affine support positive width with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
(-1, 2, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)
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.