Related but not quite what I'm looking for.
I just started learning about nets and I found that the main motivation for defining nets was the need for a unified definition of the definite integral with Riemann Sums. This answer explains it very well. So, using nets we can define the definite integral of a function $f$ on $[a, b]$ as follows:
Let $D$ be the set of all tagged partitions of the interval $[a, b]$, represented as pairs $(\Delta, \xi)$ where $\Delta$ is a finite system of points $a = x_0 < x_1 < \dots < x_{n-1} < x_n = b$ and $\xi$ is a system of intermediary points $(\xi)_{i=1}^n$ such that $\xi_i \in [x_{i-1}, x_i]$.
Define a preorder relation $\geq$ on the set $D$ for $P_1 = (\Delta_1, \xi_1)$ and $P_2 = (\Delta_2, \xi_2)$ as $P_2 \geq P_1 \iff \Delta_1 \subseteq \Delta_2$. We say that $P_2$ is a refinement of $P_1$. Then, $(D, \geq)$ is a directed set (because $\Delta_1 \cup \Delta_2$ with any suitable $\xi$ is a refinement of both $P_1$ and $P_2$)
Denote the Riemann Sums as $\sigma(f, P) = \sum_{i=1}^n f(\xi_i)(x_i – x_{i-1})$
Now, we can define integration with and without nets:
With nets: The integral $\displaystyle{{\int\limits_a^b} f(x) dx = \lim_{P\in D} \sigma(f, P)}$
Without nets: The norm of a partition $(\Delta, \xi)$ is $\displaystyle{\|\Delta\| = \max_{1\leq i \leq n} (x_i – x_{i-1})}$.
The function $f$ is integrable on $[a, b] \iff \exists L =: \displaystyle{{\int\limits_a^b} f(x)dx}$ s.t. $\forall[ (P_n)_{n\in\mathbb{N}}$ sequences of partitions from $D$, with $\displaystyle{\lim_{n\to\infty }\|\Delta_n\| = 0}]$, $\displaystyle{\lim_{n\to\infty} \sigma(f, P_n) = L}$
My question is: How can we prove that the two definitions above are equivalent (/are they?), without involving a third definition, the one with upper and lower Riemann Sums, as in the related question?
The way I tried to prove it (at least the direction "with nets" $\implies$ "without nets" which I'm more interested in) was using subnets. The definition of subnet I'm using is:
Let $(x_\alpha)_{\alpha\in A}, (y_\beta)_{\beta\in B}$ be nets. We say that $y$ is a subnet of $x$ if there exists a monotone function $h : B \to A$ such that $x_{h(\beta)} = y_\beta$ and $h(B)$ is cofinal in $A$.
Using the fact that, if all subnets of a net have a subsubnet that converges to $L$, then the net converges to $L$, it follows that if all subnets of a net converge to $L$ then the net converges to $L$ (see this question for the issue with the different definition of subnet). It's also not hard to prove the other direction.
So, since the monotone sequences (with respect to the direction on $D$) in the definition "without nets" are subnets of the net in the definition "with nets", they converge. I don't know how to go about the ones that are not monotone. I also have no idea how to go about the other implication.
Any help would be much appreciated.
Best Answer
"with sequences" ⟹ "with nets"
Prove the contrapositive by contradiction. Suppose that there is a function $f$ that is not integrable with the nets definition, but is integrable with the sequences definition with integral $L$.
So there exist an $\varepsilon>0$ such that $\forall P∈D \ \exists Q∈D$ such that $P≥Q$ and $\vertσ(f,Q)-L\vert>\varepsilon$. Let $Δ_n$ a sequence of partitions such that $\displaystyle\lim_{n\to∞}∥Δ_n∥=0$, so there exist a $E_n$, such that $Δ_n≥E_n$ and $\vertσ(f,E_n)-L\vert>\varepsilon$, for every $n$. We have that $\displaystyle\lim_{n\to∞}∥E_n∥=0$, but $\displaystyle\lim_{n\to∞}σ(f,E_n)\neq L$ a contradiction.
"with nets" ⟹ "with sequences"
Suppose that $f $ is integrable with the net definition with integral $L$. Let $Δ_n$ a sequence of partitons such that $\displaystyle\lim_{n\to∞}∥Δ_n∥=0$, and let $P$ a partition such that $\vertσ(f,P)-L\vert<\frac{\varepsilon}{2}$. Define $U_n=P\cup Δ_n$ .We have that $$ \vertσ(f,Δ_n)-L\vert<\vertσ(f,U_n)-L\vert+\vertσ(f,Δ_n)-σ(f,U_n)\vert<\frac{\epsilon}{2}+\vertσ(f,Δ_n)-σ(f,U_n)\vert $$ Because $U_n$ is a refinement of $P$.
Suppose that $P$ has $m$ points. Note that if a partition $P$ has $m$ points more that a partition $Q$, then $$ \vertσ(f,P)-σ(f,Q)\vert\leq2mK∥Q∥ $$ $K$ is a upper bound of $f$.
So $\vertσ(f,Δ_n)-σ(f,U_n)\vert\leq2mK∥Δ_n∥$.
Finally if we take a $p$ such that if $n>p$ implies that $∥Δ_n∥<\frac{\varepsilon }{4mK}$, then $$ \vertσ(f,Δ_n)-L\vert\leq\vertσ(f,U_n)-L\vert+\vertσ(f,Δ_n)-σ(f,U_n)\vert<\varepsilon\\ \forall n>p $$