A hydraulic master apparatus for a hydraulic brake or clutch of handlebar-guided vehicles having a handlebar tube includes a support, a lever pivotally connected to the support, a hydraulic cylinder sized to fit in the handlebar tube and having a piston, and a pulling device configured to exert a pulling force on the piston when the lever is actuated. The apparatus also includes a grip width adjusting apparatus, by which a distance between the support for the lever and the hydraulic cylinder is adjusted to adjust a grip width. Also included is a cover that, responsive to being connected in the handlebar tube, defines with the hydraulic cylinder in the handlebar tube a compensating chamber therebetween.