Legacy of live pursuit (7)
I believe the answer is:
bequest
'legacy' is the definition.
(bequest is a kind of legacy)
'live pursuit' is the wordplay.
'live' becomes 'be' (synonyms).
'pursuit' becomes 'quest' (synonyms).
'be'+'quest'='BEQUEST'
'of' acts as a link.
(Other definitions for bequest that I've seen before include "Something gifted in a will" , "Endowment, legacy" , "Gift left in will" , "Gift by will of personal property" , "One left in will" .)