Separate the leaves from the stems of the bok choy. Cut the stems into bite-sized chunks and shred the leaves.
Heat the olive oil in large skillet over medium heat.
Cook the bok choy stems in the oil until slightly tender, about 3 minutes; add the water and leaves and cook until the water evaporates, about 10 minutes more. Stir in the capers, garlic, and ginger; cook and stir 1 minute more. Sprinkle the vinegar and lemon juice over the bok choy and remove from heat; serve immediately.