The radioactive daughters will strongly accumulate on surfaces, even without the rain -- likely because the decay daughters are initially charged particles. Radon radioactivity is normally measured with Geiger counters

you can measure it with HPGE gamma detector by measuring its daughters Pb214 and Bi214 by 3-4 successive measurements following its decay. Also another way is with liquid scintillation using OFO scintillator by ultima gold . In every case measurements should be performed as soon as possible due to the short half life 3.8 days of radon decay

Also another way to measure radon ib rainwater is with Lucas cells.

You can bubling radon in rain water out into any other radon measuring equipments.

Thankyou so much for yours wonderful sharing scientist.

All these techniques are envisagable, but I prefer forced bubbling into continious monitor or lucas cells