The highly infectious measles virus can persist in the environment for up to two hours, health authorities say.
The case is among a string of measles infections to have occurred in Melbourne over the past year, and authorities have reminded people to check their vaccination records.
A free measles-mumps-rubella vaccine is available from GPs and some pharmacies for adults born after 1966 who do not have two documented doses of a measles-containing vaccine.
Adults born between 1966 and 1994 have been warned they may not have received full immunisation coverage against the disease, or immunity through exposure during childhood.
People who were at these locations are advised to be on alert for symptoms:
Monday, 4 February to Wednesday, 6 February (8am-8.30pm): Southbank Blvd, Southbank
Thursday, 7 February (8am-2.30pm): Southbank Blvd, Southbank
Thursday, 7 February (3pm-4pm): Skybus, Southern Cross Station to Melbourne Airport
Thursday, 7 February (3.30pm-6.15pm): Tullamarine Airport, Terminal 3
Thursday, 7 February (5.35pm-9.50pm AEST): Virgin Australia Flight VA691 to Perth