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PathsFirst Borel-Cantelli lemma

Theorem trail

First Borel-Cantelli lemma

How summable probabilities become almost-sure finite occurrence.

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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
lemma
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

If , then No independence assumption is needed.

Assumptions

  • A_n are measurable events in a probability space

Proof Sketch

The event that A_n occurs infinitely often is the intersection over N of the union of all A_n with n at least N. Apply the countable union bound to each tail union. Since the probability series is summable, the tail sums go to zero, so the limsup event has probability zero. No independence is used.

Proof Obligations

Summable probabilities for the event sequence

assumption

Verify the series of event probabilities is finite before taking tail sums to zero in the first Borel-Cantelli argument.

Without summability the first lemma gives no finite-occurrence conclusion.

Limsup as shrinking tail unions

proof step

The proof represents infinitely-often occurrence as an intersection of future tail unions and bounds each tail by the remaining probability sum.

Confusing a finite union with the tail limsup loses the almost-sure conclusion.

No independence and no converse in this direction

boundary

The first lemma does not need independence, but it also does not prove the divergent-series converse supplied by the second Borel-Cantelli lemma.

Adding independence to the first lemma or inferring the converse misstates the theorem's scope.

Tail sums must actually vanish

boundary

After applying the countable union bound to future tails, verify the remaining probability sums go to zero from summability.

Quoting Borel-Cantelli without the tail-sum limit loses the almost-sure finite-occurrence conclusion.

Source Locators

Probability: Theory and Examples (2019)

proof

Chapter 2 — Laws of large numbers and independence tools

durrett-2019:ch2

kind
chapter
review
reviewed / B
extract
manual

Chapter-level support for Borel-Cantelli lemmas.

Probability and Measure (1995)

background

Chapter 1 — Probability measure foundations

billingsley-1995:ch1

kind
chapter
review
reviewed / B
extract
manual

Probability-space foundation context.

Probability: Theory and Examples (2019)

proof

Theorem 2.3.1 — Borel-Cantelli lemma

durrett-2019:thm231

kind
theorem
review
reviewed / A
extract
manual
number
2.3.1

Anchor: Borel-Cantelli lemma

Reviewed theorem-level support for the first Borel-Cantelli lemma.

Curated Trace

  1. 1
    Tail union boundinput theorem

    Each future tail of bad events is controlled by the sum of its probabilities.

    Checkpoint

    Use the countable union bound on the tail, not only on the first N events.

  2. 2
    First Borel-Cantellilemma

    Summable probabilities force the limsup event to have probability zero.

    Checkpoint

    No independence assumption appears in this direction.

  3. 3
    Infinitely often eventinterpretation

    The lemma translates a tail-union measure calculation into finitely many occurrences almost surely.

    Checkpoint

    A_n i.o. means every tail contains some occurring event.

  4. 4
    Almost-sure convergence patterndownstream

    Many strong-law proofs reduce convergence failures to a summable sequence of tail probabilities.

    Checkpoint

    The lemma proves a sufficient condition, not a converse without independence.

Failure Modes

No independence and no converse in this direction

The first lemma does not need independence, but it also does not prove the divergent-series converse supplied by the second Borel-Cantelli lemma.

Risk: Adding independence to the first lemma or inferring the converse misstates the theorem's scope.

Check: The first lemma does not need independence, but it also does not prove the divergent-series converse supplied by the second Borel-Cantelli lemma.

Tail sums must actually vanish

After applying the countable union bound to future tails, verify the remaining probability sums go to zero from summability.

Risk: Quoting Borel-Cantelli without the tail-sum limit loses the almost-sure finite-occurrence conclusion.

Check: After applying the countable union bound to future tails, verify the remaining probability sums go to zero from summability.