gyronormed function spaces

‎in this paper‎, ‎we introduce a gyrodistance on the gyrolinear space of functions (whose gyronorm is measurable) from a measure space to the möbius disk $mathbb{d}$‎. ‎the gyrodistance in question can be expressed in terms of a modification of the lebesgue integral‎, ‎which we will call the lebesgue gyrointegral‎. ‎the gyronormed space generated in this manner‎, ‎which we will call the $l^1$ gyrospace‎, ‎is similar to the familiar $l^1$ function space in many aspects‎. ‎we establish several properties of the latter‎, ‎showing that many of them mirror those of classical $l^1$ spaces‎. ‎finally‎, ‎we show that the gyrodistance in question induces a metric topology on the $l^1$ gyrospace‎.