Yes, you can bring aerosol hairspray in your checked baggage, but there are strict quantity and safety limits set by the FAA and TSA. For "medicinal and toiletry" aerosols, the capacity of each individual container must not exceed 0.5 kg (18 ounces) or 500 ml (17 fluid ounces). Furthermore, there is a total aggregate limit: the total quantity of all such aerosols (including deodorant, hairspray, and shaving cream) cannot exceed 2 kg (70 ounces) or 2 L (68 fluid ounces) per passenger. Crucially, the release valves on the aerosol cans must be protected by a cap or other suitable means to prevent accidental discharge while the bag is being handled or subjected to pressure changes in the cargo hold. If you are carrying a standard-sized "salon" bottle of hairspray, it is perfectly fine, but "industrial-sized" containers or those without a cap may be confiscated by security during the baggage screening process.