  We consider the real-valued defocusing modified Korteweg-de Vries equation
(mKdV) on the circle. Based on the complete integrability of mKdV,
Killip-Vi\c{s}an-Zhang (2018) discovered a conserved quantity which they used
to prove low regularity a priori bounds for solutions. It has been an open
question if this conserved quantity can be used to define invariant measures
supported at fractional Sobolev regularities. Motivated by this question, we
construct probability measures supported on $H^s(\mathbb{T})$ for $0<s<1/2$
invariant under the mKdV flow. We then use the Miura transform to obtain
invariant measures for the Korteweg-de Vries equation, whose supports are
rougher than the white noise measure.
