Possibly muslin

Generally I would also say muslin. However, if it's like the bottom black dust cover, that is called cambric.

Here's a link to cambric and dust cover material for upholstery. The folks at onlinefabricstore.com have been very helpful with questions.

I’ve determined it’s cambric. JoAnn Fabrics says to get it in white order online. Thanks for your replies to my question.

The fact that you said that it is shredding makes me think of how some of these pillow forms will do that after a while. Instead of ripping as a woven fabric will, they just begin to shred fibers anywhere there is a lot of movement or abrasion.

Mostly what I see on foam cushions is a layer of batting, probalby dacron, and a non woven polyester fabric over that.

But, maybe you do have a real woven cotton cambric on there. It is not what I would have expected since you said it shredded.

Just a though.

Cambric is not cotton, nor is it woven.

Cambric was originally of linen and then cotton. It is a plain weave and tightly woven textile.

Dont believe me? Ask Mr Google. He knows everything.

It has been a while since I concerned myself with the different weaves and flavors of textiles made from natural fibers. When the term came up I had to think about it a few minutes and search through my old files in my head and I do remember cambric. But, to be certain, I did ask Mr Google just to refresh that old memory and see if it still applied.

Maybe what some are referring to as "cambric" in the modern age is some synthetic textile. My couch has some manner of synthetic textile that covers the bottom.

Maybe more high end furniture has real cambric on it, maybe not.

But, cambric IS a woven textile. Or, at least it used to be.