How to help users who ignore a push message but might want to read it later?

I have a question I'm hoping you guys can shed some light on. The mobile app I'm currently working on has a feature to send push notifications for important messages like overdrawing a bank account, bills needing to be paid, an high value item being purchased etc. Essentially messages that require immediate action or immediate awareness. There's a request to allow users to read these messages back in case they ignore them or want to read them later. I'm thinking of something like a messagebox like with e-mail, but wonder if push notifications are a suitable medium for this type of feature.

What's a suitable way to notify users of important events immediately while simultaneously allowing them to read this message back? How would this work within an app environment?