LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Basic identities for a probability measure

Statement

Let $(Ω, F, P)$ be a probability space, let $A, B \in F$ , and let $(A_{n})_{n \in N}$ be events.

$P (A^{c}) = 1 - P (A)$ .
If $A \subseteq B$ , then $P (A) \leq P (B)$ .
If $A \subseteq B$ , then $P (B ∖ A) = P (B) - P (A)$ .
$P (⋃_{n} A_{n}) \leq \sum_{n = 0}^{\infty} P (A_{n})$ , and for every natural $m$ , $P (⋃_{n < m} A_{n}) \leq \sum_{n < m} P (A_{n}) .$
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$ .
If $A_{n} ↑ A$ , then $P (A) = \sup_{n} P (A_{n}) .$ If $A_{n} ↓ A$ , then $P (A) = \inf_{n} P (A_{n}) .$

Facts & Assumptions

Given: A probability space $(Ω, F, P)$ , events $A, B$ , and an event sequence $(A_{n})$ .

[L1]

A probability measure is a measure of total mass $1$ (Probability measures and probability spaces).

[L2]

Measures are monotone, set differences subtract when the smaller set has finite measure, subadditivity holds, continuity from below holds, continuity from above holds once one set has finite measure, and finite inclusion-exclusion holds for finite-measure sets (Measures are monotone, Measure of a set difference when the smaller set has finite measure, Finite and countable subadditivity of measures, Continuity from below for measures, Continuity from above when one set has finite measure, Inclusion-exclusion for a nonempty finite family of finite-measure sets).

Proof

technique · direct

1.1L1L2

Because $P (Ω) = 1$ by [L1] and $A \subseteq Ω$ , [L2] gives $1 = P (Ω) = P (A) + P (A^{c}),$ so $P (A^{c}) = 1 - P (A)$ . Monotonicity in [L2] also gives $A \subseteq B \Rightarrow P (A) \leq P (B)$ .

1.2L1L2

Since every probability is at most $1$ , the finite-measure hypotheses in [L2] apply to $A$ , $B$ , and $A \cup B$ . Thus if $A \subseteq B$ , then $P (B ∖ A) = P (B) - P (A),$ subadditivity gives the countable and finite union bounds, finite inclusion-exclusion gives $P (A \cup B) = P (A) + P (B) - P (A \cap B),$ and continuity from below and from above give the two monotone-limit formulas, because a decreasing probability sequence always has finite first term.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 are exactly the stated probability identities.

Depends on

Used by

One-dimensional Brownian motion hits every point almost surely Corollary
One-dimensional Brownian motion is recurrent Corollary
The expectation of an indicator is the probability of the event Corollary
A divergent probability sum does not force infinitely many occurrences without independence Counterexample
Kolmogorov extension alone does not give a continuous version Counterexample
Pointwise strong laws need not give a uniform law Counterexample
The raw natural Brownian filtration need not be right-continuous Counterexample
Critical, detection, and quantification levels Definition
d-dimensional Brownian motion Definition
Brownian bridge from Brownian motion Example
Detection-limit assumptions and left-censored readings Example
Planar coordinate hitting does not imply point hitting Example
Radial second moment of multidimensional Brownian motion Example
Successive Brownian exit segments are independent copies Example
Bracketing maximal inequality for the empirical process Lemma
Independent events remain independent under complements Lemma
Logarithmic L-r covering bound for VC classes of sets Lemma
Sieve metric entropy produces uniform exponential tests Lemma
Standard-normal CDF, tail, and quantile identities for power calculations Lemma
Sub-Gaussian score bound under fixed-design incoherence Lemma
The Brownian zero set has Lebesgue measure zero Lemma
The RSK union bound localizes Plancherel profiles Lemma
The standard normal CDF has unique symmetric quantiles Lemma
Typical-projector bounds for finite quantum sources and cq ensembles Lemma
Bernstein inequality for bounded independent sums Theorem
BIC fitted-value risk bound Theorem
Bonferroni simultaneous coverage Theorem
Brownian paths are nowhere differentiable Theorem
Brownian paths are nowhere locally one-half Hölder Theorem
Brownian time inversion Theorem
Dense sample covariance concentration for sub-Gaussian vectors Theorem
Dudley entropy-integral bound for sub-Gaussian processes Theorem
Entrywise deviation for sample-centered covariance under directional Gaussian tails Theorem
Exact limits in the known-blank equal-variance normal model Theorem
Expectation factors over a finite product of mutually independent random variables Theorem
Finite sub-Gaussian maxima and epsilon-net reduction Theorem
Functional central limit theorem by bracketing entropy Theorem
Glivenko–Cantelli theorem by bracketing entropy Theorem
Hoeffding inequality for independent bounded sums Theorem
Kolmogorov zero-one law Theorem

…and 11 more results.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.1 (standard reference, not scraped)
J. R. Norris, Probability and Measure, Section 1.9 (standard reference, not scraped)