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PathsCountable union bound

Theorem trail

Countable union bound

The countable bad-event control behind almost-sure arguments.

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Research Record

A compact audit view for the theorem before the full source, proof, diagnostic, and failure-mode sections.

Claim scope
theorem
Evidence level
Exact Lean wrapper
Source precision
reviewed / B
Diagnostics
1 item
Exercises
1 exercise
Failure checks
2 checks
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Theorem Statement

For events ,

Assumptions

  • P is a probability measure
  • The events are countably indexed

Proof Sketch

Start from countable additivity on disjoint events. For overlapping events, assign each sample point in the union to the first event that contains it. Those first-hit pieces are disjoint, their union is the original union, and each piece lies inside its source event. Additivity and monotonicity give the countable union bound.

Proof Obligations

Countable events in one sigma-algebra

assumption

Verify the family is countable and every event belongs to the same probability space before invoking countable subadditivity.

Uncountable-index unions require separate measurability or covering arguments and are not justified by this theorem alone.

First-hit disjointification

proof step

The proof partitions the overlapping union into disjoint first-hit pieces, each contained in its source event, then applies countable additivity.

Skipping disjointification makes it unclear how countable additivity applies to overlapping events.

True bound may still be uninformative

boundary

The inequality remains valid even when the probability sum exceeds one, so usefulness must be checked separately from correctness.

Treating every union-bound display as an informative estimate can hide vacuous probability control.

Countability is not optional

boundary

Use this theorem for countable event families; uncountable unions need separate measurability and covering arguments before a probability bound is meaningful.

Silently replacing uncountable control with a countable union bound is a common source of invalid almost-sure arguments.

Source Locators

Probability and Measure (1995)

proof

Chapter 1 — Probability measure foundations

billingsley-1995:ch1

kind
chapter
review
reviewed / B
extract
manual

Chapter-level support for countable subadditivity and union bounds under countable additivity.

Probability: Theory and Examples (2019)

background

Chapter 1 — Measure theory background for probability

durrett-2019:ch1

kind
chapter
review
reviewed / B
extract
manual

Measure-theoretic probability background for countable unions and subadditivity.

Probability and Measure (1995)

proof

Pages 24-25 — Boole's inequality and countable subadditivity

billingsley-1995:pg24

kind
page_range
review
reviewed / B
extract
manual
pages
24-25

Anchor: Boole's inequality; countable subadditivity

Reviewed page-range support for Boole inequality and countable subadditivity.

Curated Trace

  1. 1
    Probability measurefoundation

    The theorem is countable subadditivity for a probability measure.

    Checkpoint

    All events must live in the same sigma-algebra.

  2. 2
    Countable union boundtheorem

    Overlap can only make the union smaller than the sum of event probabilities.

    Checkpoint

    The right side may exceed 1, so informativeness is separate from truth.

  3. 3
    Countable event familiesmeasure-theory use

    Almost-sure arguments require countably many bad events, not only finite families.

    Checkpoint

    Track whether the index set is finite, countable, or uncountable.

  4. 4
    Summable failuresdownstream

    Borel-Cantelli applies the countable union bound to the tail union of future events.

    Checkpoint

    Summability is the condition that makes the tail bound vanish.

Failure Modes

True bound may still be uninformative

The inequality remains valid even when the probability sum exceeds one, so usefulness must be checked separately from correctness.

Risk: Treating every union-bound display as an informative estimate can hide vacuous probability control.

Check: The inequality remains valid even when the probability sum exceeds one, so usefulness must be checked separately from correctness.

Countability is not optional

Use this theorem for countable event families; uncountable unions need separate measurability and covering arguments before a probability bound is meaningful.

Risk: Silently replacing uncountable control with a countable union bound is a common source of invalid almost-sure arguments.

Check: Use this theorem for countable event families; uncountable unions need separate measurability and covering arguments before a probability bound is meaningful.