The Extech MO280 Pinless Moisture Meter is a 5 1/2- ounce, handheld device that runs on a standard nine-volt battery like you have in smoke alarms. It uses electromagnetic technology that non-invasively measures moisture to a depth to 3/4 inches below the surface, without having to penetrate it (hence, "pinless"). An LCD readout displays the percentage of moisture in the wood or other material being tested. And hey, it comes with its own pouch.
You can download a data sheet and an owner's manual from manufacturer Extech Instruments and have your super or your building engineer take a look. The company doesn't appear to sell retail, but such companies as the pithily named Professional Equipment carry it for about $130.
Extech Instruments • 285 Bear Hill Road, Waltham, Mass., 02451-1064 • 781-890-7440