Grundbegriffe der Wahrscheinlichkeitsrechnung¶
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer.
Cited by¶
13 citations across 13 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Conditional Probability
- Conditional probability is the probability of one event \(A\) relative to the assumption that another event \(B\) is known to have occurred — formally \(P(A \mid B) = P(A \cap B) / P(B)\), defined when \(P(B) > 0\).
This sourceThe measure-theoretic axiomatization of probability, defining conditional probability as P(A∩B)/P(B) and conditional expectation with respect to a sigma-algebra.
- Conditional probability is the probability of one event \(A\) relative to the assumption that another event \(B\) is known to have occurred — formally \(P(A \mid B) = P(A \cap B) / P(B)\), defined when \(P(B) > 0\).
- Disjointness
- In set theory and probability, disjoint events satisfy P(A ∪ B) = P(A) + P(B), and partitions are families of pairwise disjoint sets that cover a whole.
This sourceAxiomatizes probability with finite/countable additivity over disjoint (mutually exclusive) events, P(A ∪ B) = P(A) + P(B), distinguishing mutual exclusivity from statistical independence.
- In set theory and probability, disjoint events satisfy P(A ∪ B) = P(A) + P(B), and partitions are families of pairwise disjoint sets that cover a whole.
- Law of Large Numbers
- The strong statement implies the weak; the weak alone permits a path that keeps returning, at ever rarer indices, to large excursions.
This sourceChapter VI treats both the ordinary and the strong law of large numbers, fixing the distinction between convergence in probability and almost-sure convergence of the empirical mean.
- The strong statement implies the weak; the weak alone permits a path that keeps returning, at ever rarer indices, to large excursions.
- Measure
- In probability a probability measure assigns total mass one to the sample space and additive masses to disjoint events — the Kolmogorov axioms are literally the measure axioms with a normalization constraint, so probability theory is, structurally, measure theory with total mass one.
This sourceAxiomatizes probability as a measure normalized to total mass one — the Kolmogorov axioms are the measure axioms (non-negativity, countable additivity) plus normalization.
- In probability a probability measure assigns total mass one to the sample space and additive masses to disjoint events — the Kolmogorov axioms are literally the measure axioms with a normalization constraint, so probability theory is, structurally, measure theory with total mass one.
- Nonparametric Methods
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Probability
- Probability is the calibrated quantification of uncertainty: a numerical assignment to events or propositions that obeys a stated set of coherence rules and supports consistent reasoning and decision-making under incomplete information, the formal commitment articulated by Kolmogorov (1933) in the measure-theoretic axiomatization of probability.
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Probability is the calibrated quantification of uncertainty: a numerical assignment to events or propositions that obeys a stated set of coherence rules and supports consistent reasoning and decision-making under incomplete information, the formal commitment articulated by Kolmogorov (1933) in the measure-theoretic axiomatization of probability.
- Set and Membership
- In statistics and probability, sample spaces and event spaces are sets, and probability measures are functions on the power set (or on a σ-algebra of measurable sets) following the Kolmogorov axiomatization
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- In statistics and probability, sample spaces and event spaces are sets, and probability measures are functions on the power set (or on a σ-algebra of measurable sets) following the Kolmogorov axiomatization
- Stationarity
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Statistical Independence
- Its precise signature is a Factorization: the joint distribution equals the product of the marginals, written \(P(A \cap B) = P(A)\,P(B)\), and in the conditional form factoring given a separator set.
This sourceGives the measure-theoretic axiomatization of probability and the definition of independence as the factorization P(A∩B)=P(A)P(B).
- Its precise signature is a Factorization: the joint distribution equals the product of the marginals, written \(P(A \cap B) = P(A)\,P(B)\), and in the conditional form factoring given a separator set.
- Stochastic Process
- In mathematics and probability theory it is the foundational object: Kolmogorov's extension theorem constructs processes from their finite-dimensional distributions, and the whole edifice — martingales, stationary processes, ergodic theory, the classification into Markov, Gaussian, Lévy, and point processes — is the study of stochastic processes and their dependence structures.
This sourceContains the extension theorem constructing a process from a consistent family of finite-dimensional distributions.
- In mathematics and probability theory it is the foundational object: Kolmogorov's extension theorem constructs processes from their finite-dimensional distributions, and the whole edifice — martingales, stationary processes, ergodic theory, the classification into Markov, Gaussian, Lévy, and point processes — is the study of stochastic processes and their dependence structures.
- Stochasticity vs. Determinism
- Kolmogorov (1933) gave this distinction its modern measure-theoretic foundation.
This sourceFounding measure-theoretic axiomatization of probability — sample space, σ-algebra of events, countably-additive probability measure, ratio definition of conditional probability — that becomes the modern mathematical substrate for the field.
- Kolmogorov (1933) gave this distinction its modern measure-theoretic foundation.
Mechanisms¶
- Finite or Countable Additivity Test
- Its distinctive contribution is fixing which regime the application requires: finite additivity (sums over finitely many disjoint pieces) or the strictly stronger countable additivity (sums over infinitely many).
This sourceDocuments finite additivity (sums over finitely many disjoint pieces); the strictly stronger countable additivity (sums over infinitely many).
- Its distinctive contribution is fixing which regime the application requires: finite additivity (sums over finitely many disjoint pieces) or the strictly stronger countable additivity (sums over infinitely many).
- Probability Measure Construction
- The three requirements are precisely Kolmogorov's axioms, which is why this construction is the standard route to a probability.
This sourceIntroduces the measure-theoretic axioms that became the standard foundation for constructing probability spaces.
- The three requirements are precisely Kolmogorov's axioms, which is why this construction is the standard route to a probability.
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Links previously used in the corpus¶
Before the registry existed this work was also linked 2 other ways.
- https://archive.org/details/foundationsofthe00kolm ×2
- https://archive.org/details/kolmogoroff-1933-grundbegriffe-der-wahrscheinlichkeitsrechnung ×2
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