Is reading an email (on a day other than Shabbat or a holiday in which melachah is forbidden) written and sent to you by someone on Shabbat or the holiday halachically permitted?
See mishna Terumot 2, 3:
המטביל כלים בשבת. שוגג ישתמש בהם מזיד לא ישתמש בהם. המעשר והמבשל בשבת שוגג יאכל מזיד לא יאכל. הנוטע בשבת שוגג יקיים מזיד יעקור. ובשביעית בין שוגג בין מזיד יעקור:
If the chilul shabbat is unintentional its allowed, ig it is intentionally send for your usage on Shabbat, it remais prohibited forever. But obviously nobody sent it for a Shabbat utility. So it is permitted.
In mishna Betsa 3, 2 we see that the permission for motsae shabbat is no immediate but after a sufficient time the mil can be written an sent.
But if the sender is non Jewish and wrote for himself (e. g. if he needs something an ask for it) you can use the mail immediately. If it was sent from a Jew for illness problem, may be that he was allowed to send and you can answer immediately when you open your email on motsash.