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Condor5
Condor5
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I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

IfLet $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}$$\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}\subseteq \mathbb R_+$ with $\mathcal A$ a directed set, and $x_{k,\alpha}\geq 0$ for each $\alpha$ and $k\in \mathbb N$, then. Then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

If $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}$ with $\mathcal A$ a directed set, and $x_{k,\alpha}\geq 0$ for each $\alpha$ and $k\in \mathbb N$, then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

Let $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}\subseteq \mathbb R_+$ with $\mathcal A$ a directed set. Then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

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Condor5
Condor5
  • 165
  • 4

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

If $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}$ with $\mathcal A$ a directed set, and $x_{k,\alpha}\geq 0$ for each $\alpha$ and $k\in \mathbb N$, then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

If $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}$ with $\mathcal A$ a directed set, then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

If $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}$ with $\mathcal A$ a directed set, and $x_{k,\alpha}\geq 0$ for each $\alpha$ and $k\in \mathbb N$, then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.

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Condor5
Condor5
  • 165
  • 4

Fatou's lemma and dominated convergence for nets and the counting measure

I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.

That is, has anyone proved the following?

If $\{x_{k,\alpha}\}_{k\in \mathbb N,\alpha \in \mathcal A}$ with $\mathcal A$ a directed set, then

$$\sum_{k=1}^{\infty} \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} x_{k,\alpha}\leq \sup_{\bar \alpha}\inf_{\alpha\geq \bar \alpha} \sum_{k=1}^{\infty} x_{k,\alpha}.$$

This result is not true if the sum were replaced by a general measure.