For a few grand............

LOL You know his wife has him in a budget.
My wife probably does not know I spent $30,000 at Digikey last year. But as long as I turn that into $130K she does not care.

 

So many variables... Detect, maybe. Able to discern signal from noise, maybe. I used to work 100W signals from Europe here in coastal Georgia on Saturday mornings until my next-door neighbor cranked up his lawnmower and trashed the band with noise every time his sparkplug fired at several hundred RPMs. I did give him a resistor plug and offered to install it for him but he declined and did it himself. Trying to get a signal out of the 80M band in the summertime is just as bad with all the atmospheric noise from lightning is just as bad if not worse. So, throw signal to noise ratio into the mix as well. The signal is probably there but can the radio circuit discriminate true signals from the noise? So many variables...

 

Can a monopole antenna detect a 1 microwatt signal? The signal strength is 1 microwatt. If so how?

Like others have stated, a signal would be measured in microvolts, not microwatts.
And before going any farther with the discussion, what frequency realm are you asking about??
And what sort of modulation, and bandwidth???
In some frequency realms the ambient noise level is in millivolts, rather than microvolts.

 

Also makes a huge difference when signal is AM, FM, SSB, CW, etc. Gotta love CW when the band conditions are iffy...