What are the halachos of wearing perftzit tzitzis? Are they permitted to wear?
Some authorities have ruled that it is permitted to wear these tzitzis (provided the beged has a shiur, which must be checked), and to recite a berachah over them (see Or Le-Zion 2:4; Rivevos Efraim 4:15).
However, others have expressed concern that the purpose of an undershirt is only to absorb sweat, and therefore it is exempt from tzitzis (see Tehceiles Ve-Argaman 5, s.v. apikresin, in the name of Rav Chaim Kanievski; Shut Chemdas Tzvi 3:3).
Shevet Mi-Yehudah (Grinwald, Ki Teitzei) writes that tzitzis must be tied specifically to an ‘important garment,’ and not to a garment that is worn on the flesh to absorb sweat. A source for this is brought from the ruling of the Rosh (Mo’ed Katan 3:43), concerning keriah, which states that an undergarment whose purpose is only to absorb sweat is not considered a garment.
[Concerning wearing one’s regular tzitzis without an undershirt, see Tzitz Eliezer 8:3.]
Yet, it does not appear that our undershirts are made solely for the purpose of absorbing sweat, for they also serve to provide a degree of warmth in the winter, and a degree of comfort in their touch. Moreover, they are sometimes worn alone, for instance when playing sport–though it stands to reason that this does not give them any great importance, for it is the way of sportsmen to wear ‘undershirts’ alone.
Therefore, it cannot be said that ‘undershirt tzitzis’ may not be worn, and those who wear them should not be told off for it, for they have poskim on whom to rely.
Yet, there is certainly a virtue in wearing the regular tzitzis (garments that are not made for the purpose of absorbing sweat), and one should definitely try to recite the berachah over regular tzitzis or over a talis.