In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.[1]
YouTube Encyclopedic
-
1/3Views:71 043448 063555 093
-
How Do Astronomers Measure Distance?
-
Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy
-
❖ The Span of a Set of Vectors ❖
Transcription
Hi, and welcome to Brain Stuff. I'm Jonathan Strickland. And I'm Ben Bowlin. And Ben, I understand we have an interesting question today. Oh, yeah. This is a good one. All right, so Jonathan. Are you ready? Yep. That's not the question. OK. All right. How do astronomers measure distance? How are they able to measure how far away a star is? Ah, right. So it turns out that measuring the distance to the star is an interesting problem. So astronomers have come up with two different techniques to estimate how far away any given star might be. Now, that first technique involves triangulation, or parallax. So the Earth orbits the sun, right? Yeah. Well, as it orbits that sun, there's a diameter of orbit about 186 million miles. So by looking at a start one day and then, maybe, six months later, looking at it again? Astronomers can see a difference in the viewing angle for that star. So with a little trigonometry-- which is, frankly, terrifying to me-- the different angles yield the distance. So this technique works for stars that are about 400 light years away from earth, or closer. Now, there's no direct method currently available to measure the distance to stars that are further than 400 light years from Earth. But there's an approximation, right? Yes. And this is a pretty clever one. This is the second technique. Astronomers can use brightness to figure out a star's distance. So they do this because it turns out that a star's color spectrum is a good indication of its actual brightness. So like blue, white, et cetera. The relationship between color and brightness has been proven using the several thousand stars that are close enough to Earth to have their distance measured directly, that would be the 400 light years and closer. So astronomers can look at a distant star, determine its color spectrum. And from the color, then they can determine the star's actual brightness. By knowing the actual brightness and comparing it to the apparent brightness seen from Earth-- that is, by looking at how dim the star has actually become once its light reaches the earth, they can determine that the distance of the star is whatever it happens to be. Right. And there's some other stuff we should probably talk about. Yeah. I've got some trivia for you, Ben. Oh, good. OK. All right, so let's talk about some of the astronomical units that astronomers use, including the astronomical unit. So the astronomical unit refers to the distance between Earth and the sun, the average distance. It's about 149 million kilometers or something. Then you've got the light-year. Yeah, what is a light-year? So, a light-year is like a regular year but with fewer calories. Heh. Actually, of course, the light year is the distance that light can travel through the vacuum of space within the span one Earth year. So that's pretty big, because light's pretty fast? Yeah, yeah. We're talking just under 10 trillion kilometers. That's a pretty long distance. But I've got something that's even longer than that. And that is a parsec. Hang on. Is this a Star Wars reference? It can be. But, all right, so, Parsec in Star Wars sounds like it's a unit of time. It's actually a unit of distance. And the way it works is you talk about the distance between our sun and some other astronomical object that has a 1-second arc degree difference in parallax angle. Now that essentially means that when we look at the star and we compare it to the sun, and we do that again at another point on the Earth's orbit, we get that 1-second arc degree. That distance actually ends up being 3.26 light-years. So it's even longer. So there you go. Astronomical units. Wait. So the Kessel Run-- We'll get to it. We'll get to it. Meanwhile, while you guys have enjoyed this video, you should show your love to us by liking it, and then subscribe to the channel. We've got a lot more questions we're going to be answering, like the Kessel Run. Thank you. So Ben, the Kessel Run is, in fact, a smuggling route in the Star Wars universe. Now, the retconned explanation of why Han Solo refers to the parsecs is that this route goes through several black holes. And for a ship to be safe, it has to go around the black holes as much as possible. So the safest route is probably around 15 parsecs. But Han Solo? He's a daredevil. And so he cuts off several parsecs and makes it more of a 12 parsec or less than 12 parsec run. Oh, that's the line. Yeah. But is that because of the Force, or something? No. Han Solo doesn't believe in the Force. He's all skill, man. It has nothing to do with some-- It's been a long time since I watched Star Trek. So I don't really, I don't know.
Properties of the trivial measure
Let μ denote the trivial measure on some measurable space (X, Σ).
- A measure ν is the trivial measure μ if and only if ν(X) = 0.
- μ is an invariant measure (and hence a quasi-invariant measure) for any measurable function f : X → X.
Suppose that X is a topological space and that Σ is the Borel σ-algebra on X.
- μ trivially satisfies the condition to be a regular measure.
- μ is never a strictly positive measure, regardless of (X, Σ), since every measurable set has zero measure.
- Since μ(X) = 0, μ is always a finite measure, and hence a locally finite measure.
- If X is a Hausdorff topological space with its Borel σ-algebra, then μ trivially satisfies the condition to be a tight measure. Hence, μ is also a Radon measure. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on X.
- If X is an infinite-dimensional Banach space with its Borel σ-algebra, then μ is the only measure on (X, Σ) that is locally finite and invariant under all translations of X. See the article There is no infinite-dimensional Lebesgue measure.
- If X is n-dimensional Euclidean space Rn with its usual σ-algebra and n-dimensional Lebesgue measure λn, μ is a singular measure with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.
References
- ^ Porter, Christopher P. (2015-04-01). "Trivial Measures are not so Trivial". Theory of Computing Systems. 56 (3): 487–512. arXiv:1503.06332. doi:10.1007/s00224-015-9614-8. ISSN 1433-0490.