Gemara

141b (Mishnah - R. Yehudah): We may even be Ma'aleh Meduma (Terumah mixed with Chulin) if there is at least 101 [volumes of mixture per volume of Terumah. We remove an amount equal to the Terumah, treat it like Terumah, and the rest is like Chulin.)

142a - Question: This is Tikun!

Answer: R. Yehudah holds like R. Shimon ben Elazar:

(Beraisa - R. Shimon ben Elazar): One may intend to be Ma'aleh Meduma from one area of the mixture and eat from the rest of it [before physically separating it].

Question (Beraisa - R. Yehudah): We may be Ma'aleh Meduma if there are 101 [in the mixture];

R. Shimon ben Elazar says, he intends to be Ma'aleh from one area, and eats from the rest. (He may not separate it!)

Answer: Indeed, R. Yehudah is more lenient than R. Shimon ben Elazar.

157a (Mishnah): We request [a Chacham or Beis Din] to permit vows, if this is for the need of Shabbos.

One may plug up a Ma'or (a hole in a wall for light);

One may measure a rag or Mikveh;

157b: Ula saw Rabah bar Rav Huna measuring a barrel of water.

Ula: The Mishnah permits measuring only for the sake of a Mitzvah!

Rabah: I measure without intent.

Rishonim

Rosh (Teshuvah 8:16): If Reuven swore to pay Shimon 100 Zuz on a certain day, and it fell on Shabbos, he gives a security on Shabbos to fulfill his oath, for this is Cheftzei Shamayim. This is no less than arranging a Shiduch for a son, for someone to teach him a trade. All the more so one may pay on Shabbos to fulfill his oath!

Tosfos (126b DH umi'Divreihem): We learn from the Heter to plug up a Ma'or to other cases. The Heter is not limited to Mitzvos. We say so only about measuring, for it is not such an Isur. It is merely Uvda d'Chol. Plugging up a hole is like Binyan. We cannot say that it was permitted for a Mitzvah.

R. Shimshon (Terumos 5:2): Rashi (Bechoros 22b DH Ta'aleh) says that "Ta'aleh" refers to Bitul, and "Tirom" refers to removing. When Terumah is Batel, one must remove [from the mixture and treat it like Terumah] lest the Kohen lose. It suffices to pay money [to a Kohen, in place of removing]. For Orlah and Kil'ayim, one need not remove. Also the Yerushalmi says that one must remove lest Kohanim lose. However, one cannot exempt himself with money. R. Eliezer and Rabanan argue only about whether or not what is removed is like pure Terumah, or only based on the calculation [of what fraction of the mixture was Terumah]. However, all agree that it gets Kedushas Terumah.

Rosh (Chulin 7:37): If Isur fell into a mixture, and before any was removed more fell in, they join to forbid it. Rabanan and R. Shimon (Terumos 5:8) argue about this regarding Terumah. R. Shimon permits only because he considers anything destined to be removed is as if it was removed, and the first Isur went away. This is only for Terumah. Regarding other Isurim, he agrees that they join. Do not say that Rabanan forbid regarding Terumah because it must be removed, therefore the first Terumah to fall was not Batel. This is wrong. One must remove only lest Kohanim lose. It does not forbid the Meduma.

Terumas ha'Deshen (54): If one did not know that an Isur became mixed until Shabbos, a Chacham may estimate whether there is 60 times as much Heter for Bitul. Seemingly, 142a shows that it is forbidden. R. Yehudah permits to be Ma'aleh Meduma. The Gemara asked that this is Tikun, and concluded that R. Yehudah is more lenient than R. Shimon [ben Elazar]. Rashi explains that since one can fulfill Oneg Shabbos without removing, one may not remove. According to Rashi and the Rosh, why is removing called Tikun? It is not needed to permit the Meduma, since one could pay money. He merely needs to estimate that there is 101 times as much Heter! We must say that this estimation is Tikun. The Gemara answered that he holds like R. Shimon. Since he could fix it without Bitul, it is not considered Tikun at all. R. Shimon forbids even the estimation, since one may fulfill Oneg Shabbos through putting his eyes on one side. This Heter applies only to Terumah, which is separated through estimation and intent. It does not apply to other Isurim. Therefore, one may not estimate it on Shabbos for Bitul.

Rejection (Terumas ha'Deshen): Rashi (141b DH Af) connotes that "Ma'aleh" refers to removing. The Gemara calls this Tikun because he must do an action. Rashi in Bechoros said that it suffices to pay the Kohen. Until he removes or pays the Kohen, we are concerned for negligence. The Rosh, and also Semag, hold similarly. R. Shimshon holds that it has Kedushah, and money does not suffice. Chachamim were more stringent about Terumah than for other Isurim, lest Kohanim lose. The Or Zaru'a says that fowl or Chayos with milk is only mid'Rabanan, therefore one may be Mevatel it in a majority. One may not do so on Shabbos, for this is the ultimate Tikun. This connotes he forbids Bitul through an action, for he discussed this above, e.g. to add water [to make a majority of Heter].

Poskim

Shulchan Aruch (OC 306:7): On Shabbos one may measure if this is needed for a Mitzvah, e.g. whether a Mikveh has 40 Sa'im.

Magen Avraham (16): The Tur (CM 73) brings the Rosh, who wrote that if Reuven swore to pay on a certain day, and it fell on Shabbos, he gives a security on Shabbos worth the full debt to fulfill his oath. The Beis Yosef said that perhaps the Rosh means that he fulfills his oath only if the security was appraised and given [to be payment] for the appraised value. R. Yerucham wrote that even if he swore to pay with coins, and not with a security, he gives a security of silver worth [the debt]. Based on this, the Shulchan Aruch (73:7) permits the appraisal, and the SMA (20) permits even measuring. (Several correct the Magen Avraham's text to say so.) This is difficult. The Rosh learned from a Kal va'Chomer. 'One may arrange a Shiduch for a daughter, or to teach his son a trade. All the more so one may pay on Shabbos to fulfill his oath!' A Kal va'Chomer cannot teach more than the source from which it is learned. In the source (arranging a Shiduch or teaching a trade) one may not mention the amount of money. The Magid Mishneh (Hilchos Shabbos 24:5) explains that the amount is not a Mitzvah, it is just saving his money [lest the other side be able to claim more than is proper]. If so, also here he must give Metaltelim that are clearly worth more than the debt, so no appraisal or measurement will be needed. Surely, one may not appraise or measure. If he has money, he should give it through a Nochri. If not, he tells a Nochri to estimate, like it says in Sanhedrin 14b, or does like in Siman 500:1. I.e. he says 'this Kli is worth as much as this', and after Shabbos they appraise it.

Mishnah Berurah (34): Even though Shevus is not permitted for the sake of a Mitzvah, measuring is different, for it is not such an Isur. It is merely Uvda d'Chol. Therefore, it is permitted for a Mitzvah.

Mishnah Berurah (35): One may measure whether a hole in a wall is a Tefach to bring Tum'ah from a room with a Mes. Even though the Kohanim could leave, since he measures a Torah Shi'ur, it does not look like Uvda d'Chol. Likewise, one may measure whether or not there are 60 times as much Heter as Isur for Bitul. Since it is for the sake of giving a ruling, it does not look like Uvda d'Chol.

Kaf ha'Chayim (62): Beis Yehudah (29) permits measuring Isurim through estimation, but not with a Kli with the Shi'ur of 60, for this is an action. Aruch ha'Shulchan (2) brought several Poskim who permit. This is only if he did not know before Shabbos that the Isur became mixed.

Kaf ha'Chayim (63): Pesach ha'Dvir permits weighing for a Mitzvah. E.g. the first day of Pesach is on Shabbos, and if he estimates enough Matzah that is surely at least a k'Zayis, he will not be able to complete the Seder, or he does not have enough Matzah. He may weigh against a fork or Peros for which he knows their weight. I am wary about this. One should weigh a k'Zayis before Yom Tov, and on Yom Tov take an amount that looks slightly larger.

Kaf ha'Chayim (64): One may measure without any purpose, just he is Misasek.

Shulchan Aruch (ibid.): One may measure the belt of a Choleh to whisper on it, like women do.

Beis Yosef (DH v'Lachash): Hagahos Maimoniyos (21:4) and the Mordechai (385) bring this Heter.

Gra (DH v'Lilchosh): The Yerushalmi says that on Shabbos we may whisper for the eye, intestines, snakes and scorpions, and we pass [Kelim] over the eye.

Mishnah Berurah (36): Refu'ah is a Mitzvah. The decree lest one grind ingredients applies only to cures with substance, e.g. a drink or food.