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Locations of foreign worship are typically of concern. You're not supposed to enter a church for this very reason.

My question relates to the idea that one finds out their home was previously used for foreign worship. Say they discovered that they live in a home that was previously used by a pagan of some sort.

Assuming that moving wasn't an option, what would be the process of purifying the space as a Jewish home? Would simply living a Jewish life and restructuring the location for Jewish worship be enough?

Clarification:

In the example I'm giving, the house itself was not built specifically for idol worship or used as a formal location of idol
worship. That being said, in the same way a Christian might light
candles at a shrine in their home, the house would have had foreign
worship occur within it. That was the context of the example I was
giving.

This is explicitly addressed in Avodah Zarah 3:7:

שְׁלשָׁה בָתִּים הֵן. בַּיִת שֶׁבָּנוּי מִתְּחִלָּה לַעֲבוֹדָה זָרָה, הֲרֵי זֶה אָסוּר. סִיְּדוֹ וְכִיְּרוֹ לַעֲבוֹדָה זָרָה וְחִדֵּשׁ, נוֹטֵל מַה שֶּׁחִדֵּשׁ. הִכְנִיס לְתוֹכָה עֲבוֹדָה זָרָה וְהוֹצִיאָהּ, הֲרֵי זֶה מֻתָּר.

There are three houses: (1) A house which was initially built for idolatry, this is forbidden. (2) It was painted or plastered for idolatry, or [otherwise] it was renewed, he removes that which is new. (3) He brought an idol into it and removed it; this is permitted.

This Mishnah is accepted at face value by the Gemara (AZ 47b only adds that if the house itself was served, it's no different than a house built for idolatry), and we accordingly pasken like it (Rambam, Hil. Avodah Zarah 8:4; Tur, YD 145; Shulchan Aruch, YD 145:3).

Therefore, in your case, where the house was not built for idolatry, merely that idol worship occurred inside it, so long as the idols are removed, the house is permissible.