A little background information is in order. Livia's parents and particularly her mother did not like Dante. He was a man who worked with his hands and was a talented craftsman. But Livia's mother expected her to marry a doctor or a professional, university educated person. In fact for her entire life, Livia's mother did not like any one in Dante's family, but had no problem using their talents as craftsmen!
Livia was a conflicted woman. On one hand she loved Dante and wanted to please him. On the other hand she wanted to keep her mother happy. To that end, she tried to improve Dante by teaching him English and later encouraging him to apply for more stable jobs so that her mom may someday approve of him. Love prevails and the people who are lied to and deceived are Livia's parents.
Letters to Livia were sent to a address of a friend that her parents didn't know. Clearly another family believed in the power of love and supported this relationship despite family objections.