I am a bit confused by what it means for a $L^p$ space to be partitioned into equivalent classes instead of functions.
I understand that give two or more functions $f$, $g$, $h,\ldots$ of which are "almost equal" i.e. differs on finite points, then the Lebesgue integral of these functions are identical and $f,g,h$ forms an equivalence class based on the relation "almost equal"
But can someone please sketch a simple picture as to what this partitioning actually look like?
For example.
What would be the entire pink blob called? What would be each of the partition be called? What are elements within each partition?
Thanks!
By "almost equal", you mean "almost everywhere equal". Which means that the two functions agree except possibly on a set of measure zero. You can kind of visualize the measure of a subset of the real line as being its "length". And in fact, this is exactly what the measure is for intervals: that is, the measure of $(a,b)$ is $b-a.$ Now, what we call measurable functions are functions that in some sense behave nicely with respect to this notion. And the graphs of measurable functions are far from easily visualized, but you can somewhat visualize what it means to be almost everywhere equal.
Picture the graph of a function (such as a nice, smooth function like $x^2$), and pick some subset of the domain that has length zero (such as a single point, or a finite collection of points, or even the entire set of rationals). If you move the graph around on these points, you won't even see any holes in the graph because you're not removing anything with width. So even though you may have changed the value of the function at quite a few points, these changes are so negligible that the graphs are indistinguishable.