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Norm convergence alone does not imply pointwise convergence

Statement

Assume the Axiom of Choice. False claim: the $L^{2}$ -norm convergence conclusion of von Neumann's theorem, by itself, yields pointwise convergence of the same sequence.

Facts & Assumptions

Given: Full choice, Lebesgue probability on $[0, 1)$ , and the half-open dyadic intervals $I_{q, k} = [k 2^{- q}, (k + 1) 2^{- q})$ for $q \geq 0$ and $0 \leq k < 2^{q}$ .

[F1]

Von Neumann gives $L^{2}$ convergence of ergodic averages (Von Neumann mean ergodic theorem in L2).

[F2]

Birkhoff's pointwise conclusion is proved using the maximal ergodic theorem, not norm convergence alone (Birkhoff pointwise ergodic theorem, Maximal ergodic theorem).

[F3]

Half-open intervals have Lebesgue measure equal to their length, and full choice is the standing assumption (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, The Axiom of Choice).

Refutation

technique · constructive norm-convergent sequence

1.1

Enumerate the functions $1_{I_{q, k}}$ level by level, listing all $2^{q}$ intervals at level $q$ before proceeding to level $q + 1$ ; call the resulting sequence $(g_{n})$ .

construct

2.1

If $g_{n}$ belongs to level $q$ , then [F3] gives $∥ g_{n} ∥_{2} = λ (I_{q, k})^{1 / 2} = 2^{- q / 2}$ . The level tends to infinity with $n$ , so $g_{n} \to 0$ in $L^{2}$ .

F3step 1.1algebra

2.2

Every $x \in [0, 1)$ lies in exactly one half-open interval at each level. Thus $g_{n} (x) = 1$ once per level and equals $0$ at all the other entries of every level $q \geq 1$ . Both values occur infinitely often, so $(g_{n} (x))$ fails to converge for every $x$ .

step 1.1algebra

3.1

Steps 2.1–2.2 show that $L^{2}$ convergence alone cannot imply full-sequence pointwise convergence. This does not contradict Birkhoff and is not offered as a counterexample among actual ergodic-average sequences: [F2] supplies their pointwise convergence by additional maximal-inequality structure.

F1F2step 2.1step 2.2discharge-construct∎

Depends on

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Sources

Omri Sarig, Lecture Notes on Ergodic Theory (standard reference, not scraped)