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Sometimes, a Lebesgue integral (see Blood, Sex, and Lebesgue Integrals) doesn’t do the trick for what we need. Mainly, Lebesgue integrals assume that your function is additive (see Fuzzy Measures). However, in analyzing real data, we want to capture the interaction between the variables (see Touching, Interaction, Parallel). So, instead we may decide to use a Choquet integral. These are, in fact, a generalization of the Lebesgue integral, just as the Lebesgue integral is a generalization of the Riemann integral. With these, you just need a monotone or efficiency measure (it can be non-additive). Like the Lebesgue integral, you integrate “sideways” (up the y-axis).
If the function to be integrated is partly negative, we can split it up into the positive and negative parts. Under the symmetric Choquet integral, you can move the negative sign from inside the integrand to outside. However, being able to do this removes the translatability, which is something that Lebesgue and Riemann integrals have. This would be important for data where the first few digits of the number are all the same, and you only want to look at the smaller changes, and then add back on the rest later—for instance, if you had the data points 10004, 10007, 10003, you might just want to look at 4, 7, and 3, and then add on the 10000 part later. This property is called transposability/ To get that with Choquet integrals (which you would then call transposable Choquet integrals), you have to get rid of the symmetric rule, but generally in real-life applications the sacrifice is worth it since there are few cases where the symmetric Choquet integral is a reasonable choice.
(Actual integral forms/equations excluded for brevity but may be requested if you are interested!)
***
“Oh,” Sherlock said, tracing his fingers over John’s stomach. “Nothing at all like the data I had been testing.”
John rolled his eyes. “You know, Sherlock, I seriously doubt my body hair is terribly different from any other bloke’s.”
“Wouldn’t know,” Sherlock said, increasing his pressure and feeling the surprisingly fine hairs bend beneath his fingers as he stroked against the grain.
“I’m sure you’ve seen plenty of peoples’ stomachs.”
“Dead bodies don’t count,” Sherlock said. “It’s different when it’s—warm. Anyway, it’s not as if I sit about and stroke corpses’ hair in the morgue.”
“Right,” John agreed. “Well, thank god for that.” He felt his face redden as Sherlock continued to feel his belly, now moving from just his fingertips to the palms of his hands as well. “Enjoying yourself?” he rolled his eyes.
“Immensely,” Sherlock said, and didn’t seem to find that even the slightest bit of an odd thing to say to one’s flatmate. He leaned closer and took in a breath, apparently attempting to inhale whatever magical qualities the hair on John’s belly possessed. “I’d say you are, too.”
“Why’s that?” Oh god, he definitely didn’t have an erection—did he? No, no, no, thank god, no, it was just Sherlock being strange again, or at least deducing from something a bit less glaringly obvious.
“You would have started complaining at least forty seconds ago if you disliked it. The threshold for neutrality is approaching—past that and I shall conclude that you enjoy this.”
John was reasonably certain that was his cue to shove Sherlock away, but he couldn’t bring himself to do it. There Sherlock was, hunched over, all but groping John’s stomach, and it would be perfectly within John’s rights to punch him in the nose for it, but of course he would never do that. Thump him in the back of his head, maybe. Anyway, it probably wasn’t the weirdest way he had ever been of assistance to Sherlock, even if it was the most—well—personal. Not that this was relating to any case, no, of course not, but then, if Sherlock could resolve this maybe he could get back to more of his usual. Still, John thought, ears reddening to match his face, he had sounded so—shy about having caught a tactile glimpse, as it were, of John’s belly days ago. It was surprising, for a man who crowded up a complete stranger’s personal space without batting an eyelash.
“Neutrality threshold passed,” Sherlock said, and he pulled back this time, staring into John’s eyes. “May I—may I continue?”
And there was Sherlock, acknowledging that he knew that John was probably enjoying this just a little more than any other flatmate would, possibly flat-out more than he strictly ought to, and giving him the option to step back, to say, yes, that’s quite enough, now if you’re not using those peaches I bought not knowing this was why I’d like to eat one now.
“S-sure,” John said. And Sherlock’s fingers touched him again, and his palms, and his head drew close, and then he was burying his face into John’s stomach. “When I asked if you were enjoying yourself—” John started, and now he was fairly certain at least half his body was bright red.
“This is different,” Sherlock said. “That was the texture experiment.”
“What’s this?”
“Mm,” Sherlock said into his belly button, and John couldn’t suppress a giggle at the little vibrations that coursed over his midsection.
“Sherlock,” John spoke through his laughter, “come on, what is it?”
“John,” he paused with his cheek against John’s stomach. “You recall what I was discussing with you before—”
“About the titles of my blog entries?”
“No, no, not that. I meant—that you contribute inexplicably to my ability to solve crimes, when we work together.”
“Oh, right, and you contribute inexplicably to the number of jumpers I have to replace every year. Well, not inexplicably.”
Sherlock turned his nose up slightly against that, which of course translated to it digging into John’s ribcage. “I was wondering if that was—usual. I mean—have your medical skills improved?”
“Only the ones involving stitching up cuts and bandaging minor injuries for particularly disagreeable patients,” he grinned. Somehow, Sherlock nestled against him like this was—okay. He glanced down. Sherlock had since moved to his knees, presumably to alleviate the sort of pain that bending over to John’s belly must have been causing his back. “But I’ll tell you what, my crime-solving skills have improved drastically.”
“Is this one of those—things? Those people things?”
“What—rubbing your friend’s belly? I don’t—”
“No, this—John, you must know what I’m talking about. It’s as if my overall functioning capacity has increased by a hundred and fifty percent.”
John was about to laugh and think up an appropriately wry response, but the statement clung to him for just a moment and—well. I might actually have an idea what he means, this time, he thought, because while John didn’t do such things as evaluate “overall functioning capacity” he certainly did notice that Sherlock had brought him to life, and did it every day. When they were on the case, when John was quibbling with him over Why Not Labeling the Experiment Broccoli As Such was Not Good, he was just—fuller. Maybe it didn’t always feel like it, because he was too busy trying to not die or trying to not kill Sherlock, but if he stepped back, he was, even now, full to bursting with—something. Maybe the foam at the top varied, if it was a good day or a bad day, a giggling-at-crime-scenes day or a Sherlock-Bloody-Holmes-if-you-do-that-one-more-time-I’ll-wring-your-neck day, but there was always—something.
“I might,” John finally said. “I think so.”
Sherlock hesitantly maneuvered his fingers around John’s sides, clasping his hands at the back in a hug around John’s waist. He rubbed his face against John’s stomach. “It’s very unusual, isn’t it? What do you think it is?”
John didn’t list any of his theories. If he knew Sherlock, Sherlock had already considered them, was already considering them, and there was nothing John could do to stop that, and nothing he could do to stop Sherlock from inevitably arriving at some conclusion—probably the correct one. Probably the same one John felt in his gut, the same one that said that this was all fine, that he could live with this, that this was right. It was certainly bizarre, but so was Sherlock studying the texture of mold in the hopes of replicating the texture of John’s stomach. Everything was bizarre. This was bizarre and pleasant. It was so—so pleasant. John closed his eyes and rested one hand on either side of Sherlock’s neck. “Dunno,” he finally said. He felt Sherlock smile against his belly.
“Can we keep it?”
“Let’s do,” John said. “Let’s give that a try.”
“John,” Sherlock said after a while, after nervous breaths and fluttering eyelashes that tickled John’s abdomen, “it makes no sense whatsoever, given my physical orientation with relation to both you and the floor, but…this is…comfortable. The most comfortable.”
“It’s not the most comfortable,” John said, and Sherlock glanced up to him with cautious eyes, wide, fearful. “I bet it’s loads better lying down on the sofa. Let’s turn on the telly, shall we?”
And they did, and John leaned back and laid on a pillow, and Sherlock stretched out with his head on John’s stomach, fingers fiddling with the bottoms of John’s trousers as he listed to John the exact order in which the police officer on the telly would investigate the various suspects. John gradually took to running his fingers through Sherlock’s hair, massaging his scalp as he went. They drifted off like that, Sherlock’s head on John’s belly, John’s fingers in Sherlock’s hair, and John woke up with a crick in his neck the next morning, but that was fine. It was all fine.