Suppose one knows almost nothing about the nature of radioactivity (like the discoverers of this phenomenon).
What are the detailed/rigorous logical steps/arguments to show experimentally that radioactive radiation ionizes air?
Have a look at the cloud chamber article in wikipedia. It shows that there is ionisation of the air.
Also the discovery of the harmful effects of radioactivity.
Here is how to build your own cloud chamber.
This sort of measurements are repeated in student labs the world over. These are the first researchers in radiation.
I would read the Wikipedia article on smoke detectors. Then I would buy a smoke detector, crack one open, and try to build a similar device myself.