Possessive me is the apparent use of "me" in places where English has "my". For example:

"That's me house" (That's my house): [ðæts mi haʊs]

This is probably not a result of confusion between the possessive pronoun "my" and the object pronoun "me", as is often believed. In Middle English

Middle English

Middle English is the stage in the history of the English language during the High and Late Middle Ages, or roughly during the four centuries between the late 11th and the late 15th century....

 my before a consonant was pronounced [mi:], like modern English me, (while me was [me:], similar to modern may) and this was shortened to [mi] or [mɪ], as the pronouns he and we are nowadays; [hi wɒz] he was; versus [ɪt wɒz hi:] it was he. As this vowel was short, it was not subject to the Great Vowel Shift

Great Vowel Shift

The Great Vowel Shift was a major change in the pronunciation of the English language that took place in England between 1350 and 1500.The Great Vowel Shift was first studied by Otto Jespersen , a Danish linguist and Anglicist, who coined the term....

, and so emerged in modern English unchanged.