Go Study with Ruby

“Thanks for helping us out Professor Luck,” you say, getting up to follow Ruby. “We really appreciate it!”

“It was good to see you,” she smiles. “You sure you don’t want to hang around a moment longer?”

“Probably not, but thanks very much!” Ruby chirps, pulling you away from the room by the hand.

You manage a wave goodbye to your professor, and then the two of you are off to the library.

Half an hour into working on the problem set in the library, you and Ruby have gotten stuck.

“I know we can construct the measure like this,” you start saying, pointing at some chicken-scratch on the blackboard in the small study room you’ve taken up, “but then…”

“But then we’re stuck,” Ruby finishes, groaning. “How is this supposed to relate to integrals? Aren’t they just backwards derivatives? Why are we going through all this nonsense?”

“And then there’s this function that refuses to converge when we do try to integrate it at the limit…” you say, staring at the picture Ruby’s sketched on the blackboard—the function is equal to 0 almost everywhere, except it’s 1 at the first k rationals in [0,1]...

“I mean it’s obviously equal to zero when k goes to infinity, right?” Ruby says, frowning. “But like, how do we prove that?”

You both stare at the blackboard in silence, lost as to how to proceed.

“Hey, you know Vignesh, right?” you suddenly say.

Ruby nods.

“He’s super smart. I bet you we could ask him to come over and work it out with us and he’d have it done in like five minutes,” you say, feeling like a real bro.

“Oh yeah, that’s a great idea!” Ruby says.

“Cool, let me just text him real quick…”

Come study with me and Ruby at the library? We’re in room 303E, you write, and the reply is instant.

Be there in five. You rock man.

“He’s going to be right here,” you say.

“Awesome. Let’s hope he can actually do this stuff,” Ruby laughs.

Five minutes later, Vignesh shows up at your room—his face is covered in sweat, his hair all slicked back with a comical amount of gel put in it, and you inwardly chuckle. You’ll have to talk to him about this, afterwards.

“Hey guys, you said you were…” Vignesh pants and places his hand on the wall, taking a deep breath. “You were working on math?”

“Yeah,” Ruby says, smiling at him. “We were working on the Borel measure questions, and then trying to work out how to construct the integral from it… I don’t know, Alex and I got confused, so don’t worry if you can’t get it either.”

“Oh, I took a brief look when she posted it, this was questions two and three, yes?” Vignesh says, his eyes lighting up.

“You got it,” you reply, passing him your printout of the questions.

His eyes scan the text for a couple seconds, and then he nods.

“Oh yeah, this is just the Lebesgue-Stieltjes integral,” Vignesh shrugs.

“The what?” you say.

“Uh… it’s just a different kind of integral. Like, you probably learned the Riemann integral in single variable calc, that sounds familiar right?”

You and Ruby both nod.

“Well it turns out that the Riemann has a bunch of problems that make it annoying to work with, things like interchanging limits, and then if you’re not working with the reals…”

He pauses, seeing that he seems to have lost both of you.

“Uh… okay, basically, the prof wants us to figure out how to construct this better integral. Look, I see you’ve got the indicator functions worked out, so that’s a good start. Now we have this Borel measure, right? So we take linear combinations of the indicator functions, and then with a couple steps…”

He scribbles down a few lines of math in a script that would put middle school girls to shame, his mu and his brackets looking straight out of a computer’s typeset.

“And there we have a working definition of the Lebesgue integral! I mean, it doesn’t work with the negatives and complex numbers yet, but the point is that we can work on the function now. See how when we Lebesgue integrate this function it just becomes equal to the measure of the rationals on [0,1]?”

He writes a couple more lines to clarify.

“And then, boom, it’s equal to 0!”

“Shit dude,” you smile, all of it suddenly making sense as he works through it.

You look over at Ruby, and see her jaw hanging open. She is very impressed with this man, and so are you.

“How did you…” she begins saying.

“We should study together more, yeah?” Vignesh grins.

“Oh yes, we definitely should,” Ruby nods.

The grin on Vignesh’s face grows wider still, and you realise the chalk in his hand has snapped in half from the pressure he’s placed on it.

“So… I’ll see you two, same time, next week,” you say, unable to keep a smile from crawling onto your face too.

Vignesh +20
Ruby +15