Is it true that measurement is purely an objective action?
and that the possibility function is collapsed due to an interaction of detectors and the measured particles?
What possibility function? There is no possibility function. You have a photon going through two slits, and a detector in the guise of a screen. The photon interacts with the detector, and you see an image on the screen.
Or is it clear today that it is really the knowledge (awareness) of the trajectory that actually determines the collapse, giving measurement and thus physical reality a fundamentaly subjective nature, as indicated by the fathers of QM?
No. Take a look at Physics World reveals its top 10 breakthroughs for 2011. It said this: “after much debate among the Physics World editorial team, this year’s honour goes to Aephraim Steinberg and colleagues from the University of Toronto in Canada for their experimental work on the fundamentals of quantum mechanics”. It also said this: “Using an emerging technique called ‘weak measurement’, the team is the first to track the average paths of single photons passing through a Young’s double-slit experiment – something that Steinberg says physicists had been ‘brainwashed’ into thinking is impossible”. The relevant article was the secret lives of photons revealed:
3D plot of a single photon showing wavelike behaviour, image from physicsworld
You can read more about it in the paper on observing the average trajectories of single photons in a two-Slit Interferometer, and in the article furtive approach rolls back the limits of quantum uncertainty. Note that the photons definitely go through both slits. That's what you'd expect, because photons have an E=hf wave nature. They aren't little billiard-ball things. They might seem that way, but they aren't. I think the best way to understand this is to say this: when you detect the photon at the screen, you perform something akin to an optical Fourier transform on it, so you convert it into something pointlike. Then when you detect the photon at one of the slits, you perform something akin to an Optical Fourier transform on it, so you convert it into something pointlike. So it goes through that slit only. So the interference patterns disappears. There are no miracles. There is no magic.
Note this in the Wikipedia delayed choice quantum eraser article: "Elementary precursors to current quantum-eraser experiments such as the "simple quantum eraser" described above have straightforward classical-wave explanations. Indeed, it could be argued that there is nothing particularly quantum about this experiment". I find the non-simple quantum eraser experiments unconvincing myself.