Lebl-Basic-Analysis-I · Problem 2.6.15

Sequences and Series

Mathematics · double-series-fubini-tonelli

Problem

(Tonelli/Fubini for sums, challenging) Suppose let {xk,ℓ}k=1,ℓ=1∞\{x_{k,\ell}\}_{k=1,\ell=1}^{\infty} denote a doubly indexed sequence and let σ:N→N2\sigma: \mathbb{N} \to \mathbb{N}^2 be a bijection. Consider the series i) ∑i=1∞xσ(i),ii) ∑k=1∞(∑ℓ=1∞xk,ℓ),iii) ∑ℓ=1∞(∑k=1∞xk,ℓ).\text{i) } \sum_{i=1}^{\infty} x_{\sigma(i)}, \qquad \text{ii) } \sum_{k=1}^{\infty} \left( \sum_{\ell=1}^{\infty} x_{k,\ell} \right), \qquad \text{iii) } \sum_{\ell=1}^{\infty} \left( \sum_{k=1}^{\infty} x_{k,\ell} \right). The expressions ii) and iii) are series of series and so we say they converge if the inner series always converges and the outer series then also converges. a) (Tonelli) Suppose xk,ℓ≥0x_{k,\ell} \geq 0 for all k,ℓk, \ell. Show that the three series i), ii), iii) either all diverge (to ∞\infty) or they all converge to the same number. In the case of divergence, some of the ``inner'' series might be infinity in which case we consider the entire sum to diverge. b) (Fubini) Suppose i) converges absolutely. Show that ii) and iii) converge and they both converge to the same number as i).

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