My advice is to look for a small or medium bag with handle if possible, and fill it with miniatures of your own products. Here I made a small list of what you need:
WIPES: Use them to remove make-up before exercise. Search for the ones without alcohol and without the antibacterial ingredient triclosan, which can irritate and dry the skin.
DEODORANT: Buy the miniature version of your choice.
BRUSH AND FLEXIBLES TO PICK THE HAIR: Keep these on hand to keep the hair off your face.
SUNSCREEN: Before leaving home, apply a good sunscreen with SPF 30 minimum. (This is a must if you exercise outdoors)
TISSUES AND HAND SANITIZERS: Every woman should have them in her purse or wallet for emergencies.