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matter; affair; case; item; subject
counter for (received) emails, text messages, voicemail messages, etc.
counter for accounts (on a website, email service, etc.)
「件」 has other readings with different meanings.
この件について説明してください。
Please explain about this matter.
その件は私が担当しています。
I'm in charge of that case.
何件かの依頼が来ていますよ。
Several requests have come in.