A New View of Analysis:

“Think about what things do rather than about how they are defined.”

That’s a nice piece of advice from Dr. Erdman in response to my question about ideals in class today. Having already taken abstract algebra, the image of an ideal in my head is as an algebraic structure with certain properties:

A subset I of a ring R is an ideal if:

I is closed under addition, and

given any element r \in R, and any element x \in I, then both rx \in I and xr \in I.

But the ideal the Dr. Erdman is talking about in section 16.2 of his text is an order ideal, and to be an order ideal, we start with a partially ordered set:

Let A be a partially ordered set, then a subset I is an order ideal of A if x \in I and y \leq x together imply that y \in I.

My question in class had to do with the rationale behind the use of the term ideal to describe two (what seem to me to be) very different things. Perhaps there was a way to make them equivalent… Perhaps if we look at a sigma-algebra as a ring under some operations, or since a Riesz space is a lattice, we get binary relations there too…

The answer is both simpler and more deep that that, let us look at what an ideal does: in abstract algebra, we came up with the idea of an ideal to allow us to create ‘cosets’ and thus have an easy way of looking at factor rings. The properties of an ideal are what is needed for it to be the kernel of some homomorphism and to then generate a homomorphism theorem for rings analogous to that for groups.

An order ideal plays a similar role in partially ordered sets, although I have not actually looked at the details of that, it makes a lot of sense. I will try to work through what I have and see where that gets us – any comments or suggestions or observations would be greatly appreciated!

We also discussed a selection of different kinds of measures:

[0, ∞] : Positive Measure
[0, ∞) : Positive Real Measure
(-∞, ∞] or [-∞,∞) : Signed Measure
\mathbb{R} : Real Measure
\mathbb{C} : Complex Measure.

In general, a measure is a countably additive set function whose domain is a \sigma-algebra, and whose codomain is some subset of \mathbb{R} (yes, or \mathbb{C}) In the list above, the first items are the codomains for that particular kind of measure.

Another thing Dr. Erdman gave us to think about was why must we have \mu(\emptyset) = 0 when we allow \infty or -\infty as a result (i.e. in either signed or positive measures?)  And then, why cannot we have the possibility the the empty set is the only set of measure 0 and all other sets have measure \infty?

There were a couple of other items I didn’t quite catch, if someone else can fill me in…  But one important thing I did catch — measures are an abstraction of the idea of length.  Perhaps more an idea of the size of a set in some sense too.

In any case, I must get cracking on class prep for my first day of teaching!

ex animo-

Felicis

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One Response

  1. Sweet! He once told me upon my questioning him about what a Tensor was, “I don’t know, but I can tell you what it does!”

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