I recently received a new gadget: The Jawbone UP band. Look it up on Google, but the basic idea is it is a bracelet you wear 24/7 that measures all kind of things while you move around.
Should it be removed on Shabbat?
TL;DR: Please do CYLOR, but one should probably remove it. I would not try to argue that it is clearly forbidden or not, though I would assert that the issues below make this complex enough that one should avoid it until they can consult with a competent halachic authority.
There are three major issues, that I can see, with wearing one of these bands, like the Jawbone UP or the Fitbit Flex: Carrying outdoors, Measuring, and Electricity (as @trying mentioned, Writing is often discussed too). Each of these things are generally forbidden on Shabbos, with some exceptions.
There are some factors that may allow one to be lenient in specific cases. For example, indirect usage can be used as a mitigating factor. @trying gives the example of opening a refrigerator, which could indirectly cause the compressor to turn on by lowering the temperature. Some authorities allow this, usually provided the effect is not immediate. It seems to me that activity bands cannot make use of this leniency, since there is no indirectness in one's activation of the sensors. They directly sense and record any movement by the wearer.
Another possibility alluded to by @trying, is the idea of "psik reisha," unintended results that were unavoidably caused by one's actions. Cases where the unintended result is also undesired are called "psik reisha d'lo nicha lei," and can be combined with other mitigating factors to allow certain activities. This is less clear, and I can see some arguing whether walking/moving/sleeping while wearing these bands would be activating the sensors "unintentionally." That seems unlikely to me: one is wearing the band for the express purpose of recording movement! Additionally, it would be difficult to argue that the resulting measurements are "lo nicha" (unwanted).
secondly: i see two possible question's:
here are some sources which may shed some light on this topic.
however regarding the issue of writing, all i can say is Making a mark (even with a fingernail) for a significant purpose is considered kosev midrabanan. but im not a rav, so i wont try to make something up.