Describing (tagged) partition.

Question

Can someone please explain what a partition is? And a tagged partition? Preferably with pictures and very few assumptions of previous knowledge.

Answer 1

You can see here in the other answer a proper and brief definition for these terms. I myself do not like the language. Usually a partition of a set $E$ is a family of subsets whose union is all of $E$ and no two members have a point in common. But, then, we use the same word for partition of an interval: a partition of an interval $[a,b]$ is now a finite family of closed subintervals whose union is all of the interval $[a,b]$ but since we cannot make them disjoint we just ask for them to be nonoverlapping. The two uses conflict but no one cares apparently.

How to obtain any partition of an interval? Well just subdivide $$a=x_0<x_1<x_2< \dots < x_{n-1} < x_n =b$$ by adding in a finite number of points $x_1$, $\dots$, $x_{n-1}$ in between $a$ and $b$ and you obtain a partition $$[x_0,x_1], \ [x_1,x_2],\ [x_2,x_3], \dots [x_{n-1},x_n].$$ The union of the intervals is all of $[a,b]$ and, while each interval contains a point in the next interval, no two intervals overlap. Unfortunately you will find some textbooks calling the subdivision itself a partition (not the intervals but rather the points).

Worse is the tagged partition. The language seems to have originated since the 1960's in studies of the Henstock-Kurzweil integral. You take each of the $n$ intervals $[x_{i-1},x_{i}]$ from the partition above and select a single point in each, say $\xi_i \in [x_{i-1},x_{i}]$. Older textbooks called them "the associated points" and many recent authors call them "tags." I prefer to just consider the collection of ordered pairs $$\{ ([x_{i-1},x_{i}], \xi_i ): i=1,2,\dots , n \}. $$ This is a collection of interval-point pairs. In the usual mathematical language this is a relation, a linking of intervals with points. In the language I learned from Federer this is a covering relation since the points are inside the associated intervals. (Advanced readers study covering relations in association with the Vitali theorem and the notions here are actually closely related to ideas one finds there.)

But as this discussion is now getting more abstract looking let me answer the question in a more proper mode. The question here probably is just

"What are these things (partitions and tagged partitions) and what do they have to do with integration theory."

Here is a mini lesson that I prefer to offer for this topic, assuming I have before me a calculus student now learning more serious integration.

If $f$ is continuous on $[a,b]$ then the integral that we (kind of) learned in the calculus has an important property called the mean-value theorem for integrals: $$\int_a^b f(x)\,dx = f(\xi)(b-a)$$ where it is possible to select a point $\xi \in [a,b]$ to make this exact. If you subdivide the interval $[a,b]$ into pieces as above $$a=x_0<x_1<x_2< \dots < x_{n-1} < x_n =b$$ then in each interval $[x_{i-1},x_i]$ from the partition you can select a single point $\xi_i$
so that $$\int_a^b f(x)\,dx = \sum_{i=1}^n \int_{x_{i-1}}^{x_i}f(x)\,dx = \sum_{i=1}^n f(\xi_i)(x_i-x_{i-1}).$$

As we see then any integral (of a continuous function anyway) can be exactly expressed as a Riemann sum over some "tagged partition" provided that you carefully choose the tags that work.

Cauchy had a similar idea. Since we don't know in advance how to choose the "tags" why not pick always the point $\xi_i=x_{i-1}$, i.e., pick the tags at the left hand endpoint of the interval. Since $f$ is continuous (but not necessarily constant) this will change the sum but maybe not by too much. Thus Cauchy would say that, approximately, $$\int_a^b f(x)\,dx \approx \sum_{i=1}^n f(x_{i-1})(x_i-x_{i-1}).$$

Riemann took Cauchy's idea one step further but chose the tags arbitrarily (not just at the endpoint and not using the mean-value theorem). According to him a function is integrable if these sums $$ \sum_{i=1}^n f(\xi_{i})(x_i-x_{i-1}) $$ are always close to some value when the partition used is "fine" enough (lots of points in the partition, and close together).

Henstock and Kurzweil took the definition one step further and came up with a more general way of describing the taggged-partitions as "fine" (not just lots of points close together but small depending also on the tags).

After all this preliminary (assuming any readers are still awake) here is the answer to your question. The study of integration theory on the real can be undertaken in many different ways. One way that is historically important as well as technically important is to relate the integral $\int_a^b f(x)\,dx$ with the values of the Riemann sums for the function $f$ $$ \sum_{i=1}^n f(\xi_{i})(x_i-x_{i-1}) $$ where we have selected points $$a=x_0<x_1<x_2< \dots < x_{n-1} < x_n =b$$ to form a partition and then in each interval $[x_{i-1},x_i]$ from the partition we have selected a single point $\xi_i$ called a "tag" (if you must!). The entire collection $$\{ ([x_{i-1},x_{i}], \xi_i ): i=1,2,\dots , n \} $$ is what is called a "tagged partition."

Answer 2

A partition of $[a,b]$ is a set of points $x_i$, with $a=x_0<x_1<\cdots<x_n=b$. A tagged partition is a partition where you add points $t_i\in [x_i, x_{i+1}]$, for each $0\le i \le n-1$.