Investigating just 50% of the problems in Greek capital (7)
I believe the answer is:
probing
'investigating' is the definition.
(probing is a kind of investigating)
'50% of the problems in greek capital' is the wordplay.
'50% of' means to take half the letters.
'capital' suggests taking the first letters.
The initial letter of 'greek' is 'g'.
'problems' cut in half is 'prob'.
'prob'+'in'+'g'='PROBING'
'just' acts as a link.
(Other definitions for probing that I've seen before include "Testing" , "Investigating - prodding" .)