{"id":"W2028178657","doi":"10.4153/cmb-2002-011-4","title":"Invariant Measures and Natural Extensions","year":2002,"lang":"en","type":"article","venue":"Canadian Mathematical Bulletin","topic":"Mathematical Dynamics and Fractals","field":"Mathematics","cited_by":15,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"","keywords":"Mathematics; Ergodic theory; Invariant (physics); Pure mathematics; Gauss; Invariant measure; Algebra over a field; Mathematical physics","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["insufficient_payload"],"consensus_categories":["insufficient_payload"],"category_scores_codex":[0.0004970196,0.0002658869,0.0004581406,0.0001559809,0.0002314101,0.0001266784,0.0002323817,0.0001589072,0.02396622],"category_scores_gemma":[0.003338904,0.0002205859,0.000105267,0.000141671,0.0001908438,0.00004094535,0.00006602649,0.000334127,0.005058986],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00007718684,"about_ca_system_score_gemma":0.00003106532,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0001818825,"about_ca_topic_score_gemma":0.0006956432,"domain_scores_codex":[0.9981812,0.00007430103,0.0004773316,0.0003401175,0.0003167535,0.0006102637],"domain_scores_gemma":[0.9975044,0.0009541565,0.00007913377,0.0004716442,0.00009343035,0.0008972797],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.000001821691,0.00008539874,0.000008449835,0.0001522693,0.00003390534,0.0001013857,0.0003639883,1.538132e-7,0.00007259224,0.9532751,0.04396614,0.00193874],"study_design_scores_gemma":[0.0003315327,0.00003513746,0.00006616665,0.0001881133,0.00006886616,0.0002825085,0.0001595957,0.01420365,0.00002197728,0.9393031,0.04488652,0.0004528348],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"empirical","genre_scores_codex":[0.3462811,0.002299753,0.02318741,0.08048072,0.0005691994,0.002963922,0.0001665986,0.0007236718,0.5433276],"genre_scores_gemma":[0.970354,0.00003937961,0.01815455,0.001264938,0.00008003176,0.000039729,0.000002990409,0.00006118196,0.01000319],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.6240729,"threshold_uncertainty_score":0.9957157,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.0417339282906584,"score_gpt":0.2432412599307847,"score_spread":0.2015073316401263,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}