presumption shifts the burden of proof

Rule: After the plaintiff produces evidence include certain reasonable presumption, the burden of proof shifts to the defendant to show the presumption is incorrect.

Application to this case: the tenant showed an email from the husband's email address, it is reasonable to presume that the husband sent the email. In order to overcome that presumption, the husband/defendant has the burden to show that the presumption is incorrect, such as showing record of hacking or unauthorized access of email system.

When you say "who can prove that email from defendant's email address was indeed sent by the defendant", you are confusing who has the burden of proof. It is not as complicated as it sounds. It is common sense. Look up "presumption" on wikipedia.