"If the self is scientifically measured, what is the Buddhist view on this?"
Let's be very clear: the self has not been scientifically measured. It has not been measured by any scientific discipline let alone social psychology which is about as soft a science as you can find. Set aside the replication crisis that is most profoundly visible in social psychology and look at your assertion on the merits. Please cite even one study or result in a peer reviewed scientific journal - regardless whether it has been replicated - that shows the objective measurement of "the self" itself.
No, you cannot do this because there exists no such study or result. Why? Because all that has been "measured" by social psychology are ideas or subjective feelings, impressions, or perceptions about the self. But ideas, feelings, impressions and perceptions are not the same as the self.
Do you stipulate this is true? I hope so, because otherwise it will be hard to go further. If not, then again please cite one study or result in a peer reviewed scientific journal that even suggests otherwise.
Assuming you do stipulate, then let's correct your assertion above to:
"If feelings, impressions, perceptions or ideas about the self are scientifically measured, what is the Buddhist view on this?"
The Buddhist view on this is that these feelings, impressions, perceptions and ideas about the self are not in dispute. Indeed, the only remarkable thing would be if social psychology found that these ideas, impressions, perceptions or feelings about the self did not exist. But of course they do. The Buddhist view stipulates that all these exist and befuddle sentient beings into believing that the self exists in a real, essential, inherent way when in fact no such real, essential, inherent self can be found.
"As there are strong evidences that the self exists, I think it's better to say "there is self, but we shouldn't think about it to avoid sufferings" rather than say "the self is an illusion"
Again, if you think there is strong evidence from social psychology that the self exists in a real, inherent, essential way then please cite even one study or result in a peer reviewed scientific journal saying so. I don't think you can. No, what you'll find is numerous findings indicating people talking about their subjective impressions, feelings, perceptions or ideas about the self. That is not scientific evidence that the self exists in a real, essential, inherent way.
BTW, I never said that the self is an illusion. Rather, it is like an illusion just as all phenomena exist in an illusory-like way. Let me say that again for emphasis: far more than just the self, all phenomena exist in an illusory-like way completely void of real, essential or inherent existence. What's more, there is not one valid empirical scientific study or result that conclusively shows otherwise. Proffer even one scientific study or result that conclusively shows otherwise and like His Holiness the Dalai Lama ... I would be more than happy to investigate and change my mind if something, anything, can be shown to exist under ultimate analysis.