Is the usage of "the a priori" correct?

I Google-Book "the a priori" signal processing and come up with at least 8 pages of definite hits. Here's one in which "the a priori" is conveniently unitalicized:

—Tirarenko, Larysa and Barkalov, Alexander, Methods of Signal Processing for Adaptive Antenna Arrays, Springer, 2013, ‘4.2 Nature of a Priori Uncertainty about Properties of Signal and Noise’

There's lots more where that came from, if Google Books gives you anything like what it gives me.

Oh, in case your editor is xenophobic, there are plenty of authors with unexceptionably Anglo names among those hits.

You don't have a usage problem; you have a typesetting problem, which is (like punctuation) a matter of style. Thus you (and your editor) should be guided by your manual of style. I use the Chicago Manual of Style, which recommends roman type for familiar foreign words; italic, for not-so-familiar. As the CMS notes:

In deciding whether or not to italicize, the author's and editor's task is to place the word on the spectrum of usage stretching form foreign-and-unfamiliar at one end to unforeign-and-familiar at the other. And for doing this there are no rules but sensitivity and common sense.

But you have an additional consideration. Typesetting (like punctuation), should encourage readers to parse your text correctly. English does not concatenate articles, so "the a" is likely to jar your readers. As a kindness to them, avail yourself of italics:

In fact the a priori reasoning is so entirely satisfactory to me that if the facts won’t fit in, why so much the worse for the facts is my feeling.