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Borel-Cantelli for the shrinking intervals $(0, 2^{- k})$ under a dyadic atomic measure

Example

On the Borel subsets of $R$ , define

$μ : = \sum_{j = 0}^{\infty} 2^{- (j + 1)} δ_{2^{- j}}, E_{k} : = (0, 2^{- k}) .$

Then $μ (E_{k}) = 2^{- (k + 1)}$ , so $\sum_{k} μ (E_{k}) = 1$ and the first Borel-Cantelli lemma gives $μ ({lim sup}_{k} E_{k}) = 0$ . In fact ${lim sup}_{k} E_{k} = \emptyset$ .

Facts & Assumptions

Given: The dyadic atomic set function $μ$ and intervals $E_{k}$ displayed above.

[L1]

Dirac set functions are probability measures (A Dirac set function is a probability measure), and nonnegative countable weighted sums of measures are measures (Nonnegative scalar multiples and countable weighted sums of measures are measures).

[L2]

The geometric series with ratio $1 / 2$ sums to $2$ , and the sequence $(2^{- k})$ tends to $0$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

[L3]

If the sum of the measures is finite, the first Borel-Cantelli lemma makes the set limsup null (The first Borel-Cantelli lemma for measures).

[L4]

The set limsup is the intersection of the tail unions (Limit superior and limit inferior of a sequence of sets), and $(0, b) = {x : 0 < x < b}$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Verification

technique · direct

1.1givenL1L4

By [L1], $μ$ is a measure. For fixed $k$ , the atom $2^{- j}$ lies in $(0, 2^{- k})$ exactly when $j > k$ , so $μ (E_{k}) = \sum_{j > k} 2^{- (j + 1)}$ .

1.2givenL2L4

The intervals decrease. No $x \leq 0$ belongs to any of them, and for $x > 0$ , convergence $2^{- k} \to 0$ supplies $k$ with $2^{- k} \leq x$ , so $x \notin E_{k}$ ; hence their intersection, and therefore their limsup, is empty.

2.1step 1.1L2algebra

The tail geometric sum in step 1.1 is $2^{- (k + 1)}$ , and therefore $\sum_{k = 0}^{\infty} μ (E_{k}) = \sum_{k = 0}^{\infty} 2^{- (k + 1)} = 1$ . At $k = 0$ , this gives $μ ((0, 1)) = 1 / 2$ .

3.1step 2.1step 1.2L3∎

Borel-Cantelli applied using step 2.1 gives $μ ({lim sup}_{k} E_{k}) = 0$ , while step 1.2 independently identifies that limsup as $\emptyset$ .

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Borel-Cantelli for the shrinking intervals (0,2−k)(0,2−k) under a dyadic atomic measure