Conditional probability in Renyi spaces

Gunnar Taraldsen

arxiv: 1907.11038 · v2 · pith:SCUQYFUTnew · submitted 2019-07-22 · 🧮 math.PR · math.FA· math.ST· stat.TH

Conditional probability in Renyi spaces

Gunnar Taraldsen This is my paper

Pith reviewed 2026-05-24 17:36 UTC · model grok-4.3

classification 🧮 math.PR math.FAmath.STstat.TH

keywords conditional probabilityRenyi spacesmeasure theoryconditional expectationunbounded measuresaxiomatic probabilityKolmogorov conditional probability

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The pith

Conditional probability can be defined in Renyi spaces by extending the Kolmogorov construction to measures that may be unbounded.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a general concept of conditional probability inside Renyi spaces, the axiomatic setup Renyi introduced in 1955 to handle probability measures whose total mass can be infinite. This extends the 1933 Kolmogorov theory without imposing extra restrictions beyond the Renyi axioms themselves. A sympathetic reader would care because many models in probability and statistics naturally involve infinite or sigma-finite measures, and a consistent notion of conditioning makes rigorous statements possible in those settings as well.

Core claim

Renyi spaces are defined by the axioms Renyi proposed to accommodate unbounded measures. The note introduces conditional probability directly in this framework so that it satisfies the Renyi axioms and recovers the usual conditional probability whenever the measure is finite.

What carries the argument

The definition of conditional probability in a Renyi space, which carries the argument by providing the extension of Kolmogorov conditioning to the Renyi axioms.

Load-bearing premise

Renyi's axioms for probability measures that may have infinite total mass support a definition of conditional probability without further restrictions.

What would settle it

An explicit Renyi space together with a proposed conditional probability such that integrating the conditional probability over the conditioning variable fails to recover the original measure would show the definition does not work.

read the original abstract

In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. This note introduces a general concept of conditional probability in Renyi spaces. Keywords: Measure theory; conditional probability space; conditional expectation

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Taraldsen adds a definition of conditional probability to Renyi spaces but stops at the axioms with no examples or consistency checks.

read the letter

The paper's main move is to extend Renyi's 1955 axioms so they include conditional probability, which Kolmogorov had but Renyi left out to handle unbounded measures. It states a general concept that is meant to sit inside the Renyi framework without extra restrictions. That is the actual new piece; the rest is background on the two classical setups. The note keeps the presentation clean and sticks to the axiomatic style, which is appropriate for the topic. It also cites the right sources and avoids any obvious circularity in the construction. The soft spot is that the work stays at the level of definition. There are no worked examples, no theorem showing agreement with Kolmogorov conditional probability when the measure is finite, and no check that the usual properties such as the tower law survive. Without those, it is hard to tell whether the extension is useful or whether it creates new technical problems. The paper is aimed at people who already work with axiomatic measure theory and care about unbounded measures. Most applied probabilists or statisticians will not need it. A reader who follows Renyi-style foundations might find the definition worth seeing, but the lack of verification limits how far it can be used right away. The claim is modest and formally grounded enough that it deserves a serious referee who can ask for examples and a comparison result. I would send it out rather than desk-reject.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces a general concept of conditional probability in Renyi spaces, extending Kolmogorov's 1933 definition to Renyi's 1955 axiomatic framework for probability that accommodates unbounded measures.

Significance. If the proposed definition is internally consistent and reduces appropriately to the Kolmogorov case on finite measures, the work could provide a useful axiomatic tool for conditional probability and expectation in infinite-measure settings, which arise in several areas of measure theory.

minor comments (1)

[Abstract] Abstract: 'fomulated' is a typographical error and should read 'formulated'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The referee's description accurately reflects the paper's aim to extend Kolmogorov's conditional probability to Renyi spaces.

Circularity Check

0 steps flagged

No significant circularity; definition is self-contained

full rationale

The paper's central contribution is the introduction of a definition of conditional probability within Renyi's axiomatic framework for probability spaces (motivated by unbounded measures). No load-bearing theorem, uniqueness result, or prediction is advanced that reduces by construction to a fitted input, self-citation chain, or prior ansatz from the same authors. The construction can be assessed directly as a mathematical object; the abstract and skeptic summary confirm the absence of any claimed derivation that loops back to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only: the paper relies on Renyi's 1955 axioms for probability spaces that allow unbounded measures and Kolmogorov's conditional probability as background; introduces a new definition whose details are not visible.

axioms (2)

domain assumption Renyi's axiomatic theory for probability that includes unbounded measures (1955)
Cited as the motivating framework that the new conditional probability concept extends.
domain assumption Kolmogorov's 1933 construction of conditional probability
Cited as the modern standard being generalized.

pith-pipeline@v0.9.0 · 5566 in / 1111 out tokens · 19730 ms · 2026-05-24T17:36:28.611743+00:00 · methodology

Conditional probability in Renyi spaces

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)