Two mathematicians, one indispensable theorem. Johann Radon (1887–1956) was an Austrian mathematician whose work quietly underpins the CT scanner that might one day save your life. Otton Nikodym (1887–1974) was a Polish mathematician — remarkably, born the very same year — who sharpened Radon's idea into the form everyone uses today.
Suppose you have two different ways of measuring "how much" — two measures. When can you
describe one as simply a re-weighting of the other? The answer, and the "density" function
that does the re-weighting, is
That innocent-looking fraction is the beating heart of modern probability: conditional expectation, likelihood ratios in statistics, and the "change of measure" that lets finance switch to a risk-neutral world all come straight from it.
Radon's other famous gift was the Radon transform, invented in 1917 as pure mathematics with no application in sight. Half a century later, engineers realised it was exactly the tool needed to reconstruct a 3D image of the body from X-rays taken at many angles — the mathematical soul of the CT scan, which won its inventors a Nobel Prize in Medicine. Nikodym, meanwhile, ended up teaching in the United States and, in his later years, wrote about the mathematics of quantum mechanics. A perfect pairing of a theorem you never see but couldn't live without.