| dc.description.abstract | This thesis is an attempt to establish an abstract model
for Lebesgue measure and Baire category.
In the introduction, we list several similarities between
Lebesgue measurable sets and sets having the property of Baire.
Then we abstract these similarities and use them as axioms.
In Chapter I, we introduce a generalized model and prove
some results that are well-known both in measure and category.
In Chapter II, we define kernels and covers. After proving
their existence for any set, we proceed to find some interesting
results.
It is very natural to consider the quotient algebra if
we have an algebra containing a proper ideal. Hence Chapter III
inevitably comes into the scene.
In Chapter IV we introduce analytic sets through A-operations.
This approach enables us to prove that every analytic set
belongs to our model.
In Chapter V we consider the local properties of sets
and prove some interesting results.
Chapter VI is taken from the work of J. C. Morgan II.
We include his work here for the completeness of the thesis. Also,
as we will see, it gives us a new insight into "negligible sets".
We conclude the thesis by setting up a list of questions
which, we think, are rather challenging. | |
