Option drag a copy of the shape you want add texture to
Select the Gradient Tool and fill the shape with a gradient.
One end is 50% black with an opacity of 100%.
The other end is white with an opacity of 0%.
Go to Effect > Effect Gallery...
Select Grain and Stippled.
Play with Intensity and Contrast.
Click OK
Go Object > Rasterize > Bitmap at 300ppi
Live trace with the settings we have used before.
Then Expand.
Recolour and drag back over the original shape.