When a laboratory analyzes a sample for radioactivity, it has to compare the instrument's sample results with the instrument's background results.
A one time measurement of the radiation background may or may not represent the true background, there is variability. Same with the sample.
If you've ever read the results from a laboratory, when the sample and background counts are the same, what do they record as the sample's radioactivity?
Hint: It's not zero!
In other words, if the background count rate is 60 counts per minute and once a sample is counted the lab gets 60 counts per minute, what is the radioactivity of the sample?
Sample Activity = Sample Count Rate - Background Count Rate
In this case, the laboratory would record something like "<MDA", which means "less than the Minimum Detectable Activity".
Since both the background and sample count rates are variable, the instrument is limited in its ability to resolve differences between the sample and background. There may be some difference, but the instrument is incapable of resolving it. Counting the background and sample longer could help. Using a more sensitive instrument could help, though all instruments are limited.
This is analogous to the theory of LNT. Using epidemiology, we can only resolve to a limited dose, the difference between an exposed group of people (sample) and an unexposed group of people (background). If at a certain dose we find no difference in cancer incidence, that doesn't mean there is no difference. Our ability to resolve any difference is limited. We still know that radiation and DNA have the same fundamental properties above and below that dose resolution level.
So we infer (not assume!) there is still an undetectable increase in cancer.
If you can accept <MDA (I've never met anyone who couldn't), you should be able to accept LNT. If you can't accept it, then you have a bias that you'll need to try to overcome.