  We study the radiative leptonic decays
$P\to\ell\nu_\ell\,\ell^{\prime\,+}\ell^{\prime\,-}$, where $P$ is a
pseudoscalar meson and $\ell$ and $\ell^\prime$ are charged leptons. In such
decays the emitted photon is off-shell and, in addition to the "point-like"
contribution in which the virtual photon is emitted either from the lepton or
the meson treated as a point-like particle, four structure-dependent (SD) form
factors contribute to the amplitude. We present a strategy for the extraction
of the SD form factors and implement it in an exploratory lattice computation
of the decay rates for the four channels of kaon decays
($\ell,\ell^\prime=e,\mu$). It is the SD form factors which describe the
interaction between the virtual photon and the internal hadronic structure of
the decaying meson, and in our procedure we separate the SD and point-like
contributions to the amplitudes. We demonstrate that the form factors can be
extracted with good precision and, in spite of the unphysical quark masses used
in our simulation ($m_\pi\simeq 320\,$MeV and $m_K\simeq 530\,$MeV), the
results for the decay rates are in reasonable semiquantitative agreement with
experimental data (for the channels where these exist). Following this
preparatory work, the emphasis of our future work will be on obtaining results
at physical quark masses and on the control of the systematic uncertainties
associated with discretisation and finite-volume errors.
