On August 24th, 2018 I gave a short talk at the KNAW Academy Colloquium on Generalised Baire Spaces taking place August 23rd and 24th in Amsterdam, The Netherlands.

*Abstract:* Lebesgue introduced a notion of density point of a set of
reals and proved that any Borel set of reals has the density
property, i.e. it is equal to the set of its density points up to a
null set. We introduce alternative definitions of density points in
Cantor space (or Baire space) which coincide with the usual
definition of density points for the uniform measure on
${}^{\omega}2$ up to a set of measure $0$, and which depend only on
the ideal of measure $0$ sets but not on the measure itself. This
allows us to define the density property for the ideals associated to
tree forcings analogous to the Lebesgue density theorem for the
uniform measure on ${}^{\omega}2$. The main results show that among
the ideals associated to well-known tree forcings, the density
property holds for all such ccc forcings and fails for the remaining
forcings. In fact we introduce the notion of being stem-linked and
show that every stem-linked tree forcing has the density property.

This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.

Slides are available here.