How do signatures or MACs achieve authenticity, integrity or non-repudiation？
The 3 properties have commonly accepted meaning in Cryptography: 


Authenticity: The message comes from the party associated with the verification key.
Integrity: The message had not been modified.
Non-repudiation: The signer cannot deny themselves signing the message.


We usually model the security against the weakest point of the signature/MAC - being forged; whereas other types of attacks such as key-recovery automatically implies the algorithm is insecure. 

The unforgeability implies the other 3 (2 in case of MAC) properties, because the ability to forge a signature/MAC under a particular algorithm implies that 


the message may come from someone capable of forging the signagure/MAC (authenticity), 
capable of modifying a signed message (integrity), and 
capable of signing messages against the wish of the holder of the private key (non-repudiation).