As is well known, a Jew who bought or received an eating utensil or receptacle from a non-Jew may not use it for food until it has been immersed in a mikveh. If one did not immerse it before Shabbat, some maintain that doing so on Shabbat is forbidden because it looks like one is fixing the implement (tikun kli). After all, before its immersion, one may not use it, while afterward, one may (Rosh). Others maintain that if one needs to eat from it, he may immerse it on Shabbat with a berakha. Accordingly, immersion is not considered fixing, because if be-di’avad the kli was used for food without having been immersed, the food retains its kosher status (Rif). If there is a trustworthy non-Jew in the vicinity, it is proper to give the kli to him as a gift, and then request his permission to use it. In this way, a Jew may use the kli for food even though it has not been immersed (SA 323:7). After Shabbat, it is proper to ask the non-Jew to give it back to him as a gift; he should then immerse it with a berakha.
All agree that one may immerse on Shabbat to purify himself from tum’a. Even those who maintain that kelim may not be immersed because it resembles fixing them agree that when it comes to people, since one may bathe, and since the immersion will not necessarily be seen as an act of purification, one may also immerse to purify himself (see above 14:9). In contrast, one converting to Judaism may not immerse on Shabbat as part of the conversion process because his immersion is transformative; he becomes a new person, a process that certainly qualifies as a tikun. Additionally, immersion for conversion requires the presence of a rabbinic court; just as a court does not meet on Shabbat for judgment, it also does not meet to supervise immersion (Yevamot 46b). If a court transgressed and did supervise an immersion on Shabbat, the immersion is valid, and the person is Jewish (SA YD 268:4).
One may not measure anything on Shabbat, because measuring is a weekday activity (SA 306:7; MB ad loc. 34). Therefore, one may not weigh himself or measure his height on Shabbat (SSK 14:42). Similarly, one may not measure the dimensions of furniture or a room.
One may measure and weigh for the sake of a mitzva. Therefore, one may check whether a mikveh has sufficient water (forty se’ah). One may also measure out medicine for sick people and take their temperature (SA 306:7; SSK 40:2). Since a baby has the same halakhic status as a sick person, when necessary one may measure out the amount of food that a baby needs. Similarly, when necessary one may measure if a baby has gained weight after eating (using a non-electric scale; SSK 37:5).