My brother thinks my email has been hacked. How do I check this?

Probably the first step would be to ask brother what makes him think that?

Most of us find at some time that our email address is appearing in the "from:" field of spam email messages. Often we learn of this through "bounces", messages that were undeliverable and hence returned to the "from:" address. In virtually all such cases, it is simply a spammer using your address in his messages. (Moral: be careful where you post your address.) Whilst Thunderbird makes it difficult to do this, there is a lot of software that does allow email messages to be sent using stolen or made up email addresses. But spammers need working email addresses to send to, so they'll dip into their pool of addresses, usually bought on a shady website, and use some of them as "from:" addresses. There is a small chance that some of their spam will fool someone into thinking it's really from you, or whoever else's email address is being used this week.

It's quite rare for your address being used as "from:" to mean that your account has been hacked. If your address has been harvested and is being used by spammers, then it may occur to you to stop using the account and set up a new one, though of course this is disruptive and can play havoc with all those services which use that emaii address to identify and authenticate you.

I'd say change your account's password, as a precaution, and soldier on. Eventually they will move on and stop using your address.