No offence but I don't think Lady Gaga innovated anything. The electro-dance pop sound was already trending by the time she released The Fame (just the previous year we had Rihanna's Don't Stop The Music and Disturbia, Kylie Minogue's X album, Flo Rida's Low, Hilary Duff's Dignity album, M.I.A.'s Paper Planes. Rihanna's SOS was already 3 years old by then. The Dr. Luke/Max Martin/Bloodshy & Avant electro/rock/pop/hip hop sound was already in full swing after Kelly Clarkson's Since U Been Gone, Avril Lavigne's Girlfriend, The Veronicas 4Ever, Britney's Blackout and Circus albums. The tropical pop vibes of Eh Eh Nothing Else I Can Say had Kevin Lyttle's Turn Me On and Paris Hilton's Stars Are Blind as a blueprint. Musically, The Fame gave us nothing that hadn't come out before, just now with performance art lifted from David Bowie, Freddie Mercury, Elton John, and modern pop-art artists.
Or if we want to talk about how Alejandro's reggae pop was a callback to Ace of Base, Born This Way was Express Yourself 2.0, The Fashion of His Love was I Wanna Dance With Somebody 2011, You & I was a Shania Twain song + the step step clap of We Will Rock You... I can keep going.