The purpose of integrity is to ensure that the message arrives uncorrupted, or at least that any corruptions will be detected. To achieve this, the integrity system must add redundant information to the message. This increases its size. The purpose of compression is to reduce the size of the message so that it uses up less disk space, or network bandwidth.
Compression reduces the size of the representation of the message.
Encryption usually does not change the size of the message. These three technologies are fundamental to all data manipulation and have been deployed in various forms since the advent of computers. These three technologies are tightly bound together and reinforce each other in synergestic ways. For example, compressing a message before encrypting it makes it harder to decrypt because the compression reduces the sparsity of the set of recognizably-valid decrypted messages. Similarly, integrity techniques makes compression possible because without error checking, the redundancy-reducing properties of compression would amplify transmission errors.