# [FOM] Measure and category

Tomek Bartoszynski tomek.bartoszynski at gmail.com
Sun Sep 23 19:58:26 EDT 2007

```There is a large number of results addressing the connections and
strength of measure vs category assumptions and many seem to indicate
that measure is "stronger" than category.  Many of them are contained
in the book "Set Theory: On the structure of the real line" by H.
Judah and myself, which for the most part deals with measure and
category in the set theoretic context.  There is quite a bit more
since the book was published over ten years ago.

Perhaps the most fundamental of these results is as follows.

Suppose that P and Q are partial orders. We say that P < Q if there is a
mapping from P into Q such that preimage of every bounded set (in sense of
P)  is bounded (in sense of Q).  Bounded means here "below a single
element". Such mapping is called Tukey embedding.
Let M and N be the partial orders of first category or measure zero
sets ordered by inclusion.
We have M < N.
One of the consequences of this fact is, for example, that if a union
of any kappa null sets is null then the same holds for category.
Needles to say, and this is what breaks the symmetry, N < M does not
hold.

--tomek bartoszynski