Measure of interest concerned with part of stage (5)
I believe the answer is:
apron
'part of stage' is the definition.
(I've seen this before)
'measure of interest concerned' is the wordplay.
'measure of interest' becomes 'apr' (I can't explain this - if you can you should give a lot more credence to this answer).
'concerned' becomes 'on' (I've seen this in other clues).
'apr'+'on'='APRON'
'with' acts as a link.
Can you help me to learn more?
(Other definitions for apron that I've seen before include "Protective garment worn at front" , "Garment for protection" , "Airport area for manoeuvring and loading planes" , "something befitting kitchen-sink drama?" , "Protective garment - part of stage" .)