Here's an example where visualizing things can help with problems in analysis:

Let E be a nonmeasurable set in R. Show that there is some δ>0 such that for all measurable sets A,B such that
A⊆E⊆B, we have m(B\\A)≥ δ.

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p.s. I CANNOT draw well

At first it looked scary to me because nonmeasurable sets are nasty. And in fact, this problem sort of illustrates how nasty a nonmeasurable set can be.

Let's look at what's happening.

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E is the blue oval, A is the red one, B is the green one. m(B\\A) is the area of the pink region.

What the problem says is that, there is a limit to how small the area of the pink region is.

And isn't that weird?

(3/16)

Because by making A "wider" and B "narrower" you'd think that you can shrink the pink region to as small as you want.

This leads us to think that it is the non measurability of E that breaks things. So, what do we do?

(4/16)

Well, let's see what happens if we can actually SHRINK the pink region to something of zero area.

And how do we do that? By choosing "larger and larger" A's and "smaller and smaller" B's.

They kinda look like this:

(5/16)

What happens is, you get a sequence of sets A_n and B_n such that:

1. A_n⊆E⊆B_n
2. The area between B_n and A_n goes to zero. (This is because, we ASSUMED that we can shrink the OG pink region indefinitely)

Note: The sets need not be MONOTONE.

(6/16)

I just drew them to be monotone because it is easier to draw

Now, since we assumed the negation of the conclusion, we are seeking a contradiction.

And what did we say before that might be the reason why things are "pathological"?

E is nonmeasurable.

(7/16)

So perhaps the contradiction would be that E will be MEASURABLE.

How will we show this? Let's look at the picture again.

Here I separated the OG pic to two situations:

(8/16)

One, we can make A_n's "closer" to E. The other one, we have B_n's "closer" to E.

Now, we know from measure theory that measurable sets are "well-behaved" in the sense that if you take countable unions of them, what you get is still measurable.

(9/16)

And how does this help? Well,

1. The A_n's and B_n's are measurable

2. We can "approximate" E with the UNION of A_n's or INTERSECTION of B_n's (you can see these in the pictures!)

So, say we take the union of A_n's to be "approximation" of E.

(10/16)

One can show that

m(E \\ finite union of A_n)->0.

And hence, one can show that m(E \\ union of all A_n)=0

Measure theory tells us that E\\∪A_n must be MEASURABLE.

Thus,

E = (∪A_n) ∪ (E \\ ∪A_n),

i.e., E is the union of 2 measurable sets, and must be measurable.

(11/16)

That is the contradiction!

Note that the statements in this thread can be made precise but this is the general idea.

I've been seeing a lot of discussions on how bad analysis is taught and I think a lot of the technical details obfuscate the essence of things.

(12/16)

Rigor is important.

But, I also think it's important to teach students how to visualize things. That the things they study relate to things they already know. Like how measure can be a tricky thing but for "nicer" sets, it's just their usual length/area/volume.

(13/16)

And that, barring pathological examples, a lot of what they are familiar with would hold true in these abstract spaces.

But to make things precise, one needs to put in the work learning rigor. You'll get to see how powerful it can be by going through the details.

(14/16)

Some of the analysis courses I've taken had the reputation, at least to our local Math community, that you need to memorize all the proofs. Professors didn't quite motivate the tools we use, the objects we study, or how they all relate to each other :(

(15/16)

Analysis is really beautiful, at least to me. Maybe a pedagogical shift is due to make it more welcoming to everyone.

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