  For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the
Hardy-Littlewood maximal-function of $f$ is given by the following
`supremum-norm':
$$Mf(x):=\sup_{r>0}\frac{1}{\mu(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|d\mu.$$
In this note, we replace the supremum-norm on parameters $r$ by
$\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define
Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$
converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear
operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$
is a Lie group, $d$ is a left-invariant metric, and $\mu$ is a left
Haar-measure (resp. right Haar-measure) are studied.
