Credits: 10 |
Convenor: Will Light |
Semester: 2 |

Prerequisites: |
essential: MC144,MC145,MC146,MC147,MC248,MC241 | |

Assessment: |
Continual assessment: 20% | 1.5 hour exam in January: 80% |

Lectures: |
18 | Classes: |
none |

Tutorials: |
6 | Private Study: |
51 |

Labs: |
none | Seminars: |
none |

Project: |
none | Other: |
none |

Total: |
75 |

Let's start with continuity. Suppose we have a function *f* mapping some sort
of
objects (which we will call `points') to numbers. What does it mean for the
function to be continuous at the point *p*? Well, roughly speaking, if *q* is
close to *p*, then *f*(*q*) must be close to *f*(*p*). OK so far? Well, not really.
We've already been much too loose in the way we have formulated the problem.
First off, we described the things on which *f* acts as `objects', and then
used
the more emotive word `points'. Then we went on to talk about these `objects'
or
`points' being close together. This places some restriction on the sort of
objects allowed. For example, our set of objects could consist of all cheeses
currently lying on the shelves of Sainsbury's supermarket. For a particular
cheese *p*, *f*(*p*) (which remember has to be a real number) could be the weight
of the cheese. Now I think you're going to have a hard job coming up with an
appropriate definition of two cheeses being close together! Note here that the
problem in determining a sensible meaning for `closeness' only occurs in the
input space, or the domain of the function - the set of all cheeses in
Sainsbury's. Closeness in the output space is easy to define isn't it? We would
say that *f*(*p*) is close to *f*(*q*) if |*f*(*p*)-*f*(*q*)| is small. But hang on a
minute. Even this idea is not precise enough for a mathematician (or even for
an applied scientist, or person working is some sort of technological
industry).
If I was to discover that the north wall of my house was 2mm higher than the
south wall, I wouldn't exactly be rushing to sue the builder for negligence. On
the other hand, if the garage sets the gap between my spark plug electrodes
with
an error of 2mm, then I would be bleating in their ears that something is wrong
with my car. So the notion of closeness needs to be expressed a lot more
carefully. (The formalisation of the concept of closeness in leads
inexorably via MC146 to the *formal* definition of continuity using
and technology.) But around the turn of the century,
mathematicians wanted to get away from the idea of `closeness'. This desire was
bound up with another very sophisticated development in thinking, which has
today become second nature to professional mathematicians. Traditionally,
functions used to be things which mapped numbers to numbers, or more generally,
`points' in to `points' in . But it became increasingly common
to
think of functions in a much more abstract way. In the vanguard of this
development was the idea that the so-called `points' could themselves be
functions. For example, the `points' could consist of all real-valued functions
differentiable on the whole real line. Each `point' is now a function from
to . Thus the function defined by the rule
*f*(*x*)=*x ^{2}* would be a `point'. Now consider the mapping called
`differentiation'. Applying this mapping to the `point'

Of course, we cannot in this course be anarchists, and develop our own notion which fails to coincide with the MC146 notion of continuity. So there are constraints.

Let's go on now to talk about the other great concept of integration. At the
outset, the `A'-level student knows that integration has two meanings -
definite integration (which is the area under the curve), and indefinite
integration (which is the reverse process to differentiation). Of course, the
two are linked through the fundamental theorem of calculus. On a personal note,
it's one of the great surprises to me, and something that I find absolutely
fascinating, that the area under a curve *f* between *a* and *b* can be
evaluated if only you know a function *g* such that *g*'=*f*. One version of the
fundamental theorem is then that

The astute reader will pick up on a significant difference between the first question about continuity and the second about integrability. In our continuity challenge, we allowed both the domain and the range of the function to be arbitrary. In our integration challenge, we allow the domain to be arbitrary but restrict the range to be in the friendly old real numbers . It is in the nature of things that integration requires this, after all, we want to be a real number!

- be able to define a topology, and say what is meant by a continuous function in a topological sense;
- be able to test in easy cases whether a function is continuous;
- have the notion of a compact set, and the basic theorem about the attainment of sup and inf for a real-valued function;
- be able to define a measure space, a measurable function, and the integral of simple functions and measurable functions, where appropriate;
- be able to test in easy cases whether a function is measurable;
- be able to verify the elementary properties of the Lebesgue integral;
- have seen a selection of the results about the behaviour of the Lebesgue integral under limit processes.

The ability to apply taught concepts to new situations.

The ability to write mathematics concisely.

The ability to construct logical arguments.

The ability to carry out elementary manipulations with unions and intersections of sets.

Conditions for to be continuous/measurable.

smallest -algebra containing a collection. How to construct a measure on a topological space, Lebesgue measure on . measurable for all implies

closed sets in a topological space.

subspace topology,

Compactness, continuous image of compact space is compact, closed subset of compact space is compact.

Hausdorff spaces, compact subset of a Hausdorff space is closed.

Heine-Borel theorem

Set in compact iff closed and bounded. Real-valued function attains its bounds on compact set.

sup and lim sup are measurable. is measurable. lim of every pw convergent sequence of measurable functions is measurable.

simple functions. Exists such that for all measurable

Integration of positive function via simple functions, elementary properties of the integral. and defines a measure.

Lebesgue monotone convergence theorem.

**W. Rudin**,
*Real and Complex Analysis*,
McGraw Hill, 1970.

**W. A. Sutherland**,
*Introduction to metric and topological spaces*,
Oxford : Clarendon Press, 1975.

**W. Light**,
*Introduction to Abstract Analysis*,
Chapman and Hall, 1990.