A (
$d$
-parameter) basic nilsequence is a sequence of the form
$\psi (n)= f({a}^{n} x)$
,
$n\in { \mathbb{Z} }^{d} $
, where
$x$
is a point of a compact nilmanifold
$X$
,
$a$
is a translation on
$X$
, and
$f\in C(X)$
; a nilsequence is a uniform limit of basic nilsequences. If
$X= G/ \Gamma $
is a compact nilmanifold,
$Y$
is a subnilmanifold of
$X$
,
$\mathop{(g(n))}\nolimits_{n\in { \mathbb{Z} }^{d} } $
is a polynomial sequence in
$G$
, and
$f\in C(X)$
, we show that the sequence
$\phi (n)= \int \nolimits \nolimits_{g(n)Y} f$
is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system
$(W, \mathcal{B} , \mu , T)$
, polynomials
${p}_{1} , \ldots , {p}_{k} : { \mathbb{Z} }^{d} \longrightarrow \mathbb{Z} $
, and sets
${A}_{1} , \ldots , {A}_{k} \in \mathcal{B} $
, the sequence
$\phi (n)= \mu ({T}^{{p}_{1} (n)} {A}_{1} \cap \cdots \cap {T}^{{p}_{k} (n)} {A}_{k} )$
,
$n\in { \mathbb{Z} }^{d} $
, is the sum of a nilsequence and a null-sequence.