  The standard $\Lambda$-CDM predicts a growth of structures which tends to be
higher than the values of redshift space distortion (RSD) measurements, if the
cosmological parameters are fixed by the CMB data. In this paper we point out
that this discrepancy can be resolved/understood if we assume that the graviton
has a small but non-zero mass. In the context of the Minimal Theory of Massive
Gravity (MTMG), due to infrared Lorentz violations measurable only at present
cosmological scales, the graviton acquires a mass without being haunted by
unwanted extra degrees of freedom. While the so-called self-accelerating branch
of cosmological solutions in MTMG has the same phenomenology for the background
as well as the scalar- and vector-type linear perturbations as the
$\Lambda$-CDM in General Relativity (GR), it is possible to choose another
branch so that the background is the same as that in GR but the evolution of
matter perturbations gets modified by the graviton mass. On studying the fit of
such modified dynamics to the above-mentioned RSD measurements, we find that
the $\Lambda$-CDM model is less probable than MTMG by two orders of magnitude.
With the help of the cross correlation between the integrated Sachs-Wolfe (ISW)
effect and the large scale structure (LSS), the data also pin-down the graviton
mass squared around $\mu^2\approx - (3\times 10^{-33} \rm{eV})^2$, which is
consistent with the latest bound $|\mu^2|<(1.2\times 10^{-22} \rm{eV})^2$ set
by the recent LIGO observation.
