Let
E
E
be a metric compact space. We consider the space
K
(
E
)
\mathcal {K}(E)
of all compact subsets of
E
E
endowed with the topology of the Hausdorff metric and the space
M
(
E
)
\mathcal {M}(E)
of all positive measures on
E
E
endowed with its natural
w
∗
{w^{\ast }}
-topology. We study
σ
\sigma
-ideals of
K
(
E
)
\mathcal {K}(E)
of the form
I
=
I
P
=
{
K
∈
K
(
E
)
:
μ
(
K
)
=
0
,
∀
μ
∈
P
}
I = {I_P} = \{ K \in \mathcal {K}(E):\mu (K) = 0,\;\forall \mu \in P\}
where
P
P
is a given family of positive measures on
E
E
. If
M
M
is the maximal family such that
I
=
I
M
I = {I_M}
, then
M
M
is a band. We prove that several descriptive properties of
I
I
: being Borel, and having a Borel basis, having a Borel polarity-basis, can be expressed by properties of the band
M
M
or of the orthogonal band
M
′
M’
.