 Solution

For a lower semicontinuous function, each upper-ray inverse image is open, hence Borel, so part (a)(i) applies directly. For an upper semicontinuous $h$, the function $-h$ is lower semicontinuous becauseis open. Negation is continuous and Borel measurable, so $h=-(-h)$ is Borel measurable too. Thus both semicontinuity classes are Borel measurable.