A filter in $\mathbb{R}$ by means of the Riemann sums

Question

I trying to find a filter $\mathcal{F}$ in $\mathbb{R}$ defined by means of the Riemann sums. In the other words, let $f: [0,1]\subset \mathbb{R} \longrightarrow \mathbb{R}$ be a continuous function. Consider the Riemann sum, for $r \in \mathbb{N}$, $$ S_r:=\sum_{j=0}^{r}(t_i-t_{j+1})f(\tau_j),$$ where $0=t_0<t_1<t_2<\cdots<t_j<\cdots<t_{r+1}=1$ and $t_j\leq \tau_j \leq t_{j+1}$, for each $j=0,1,\cdots, r$.

I thought of defining the family $$\mathcal{F}:=\{A_r \subset \mathbb{R} \; ; \; r \in \mathbb{N}\},$$ where for each $r \in \mathbb{N}$, $A_r:=\{S_1,S_2,\cdots,S_{r-1}\}.$

But I was unable to show that this family is a filter. Is it another family that will form the sought filter?

Because I haven't studied (and I'm not studying) the concept of net. — Guilherme, May 21 '20 at 15:33
You should for the case of Riemann sums, it’s very natural there. Kelley uses it as an important example in his book on General topology. And since then analysis researchers have embraced nets as a tool. Nowadays it seems filters are gaining traction again ( maybe due to Terence Tao?). It’s good to know about both. There are translations from nets to filters and back. — Henno Brandsma, May 21 '20 at 16:02
Or this paper the status of which is unclear to me.. just Googling. — Henno Brandsma, May 21 '20 at 16:22

score 2 · Accepted Answer · answered May 21 '20 at 16:37

2

This introduction introduces and motivates both nets and filters, and ends with how the Riemann integrals are the limit of certain net.

A filter can then be found by taking the "filter of tails" corresponding to the aforementioned net.

answered May 21 '20 at 16:37

Henno Brandsma

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A filter in $\mathbb{R}$ by means of the Riemann sums

1 Answers1